On the number of ॲ
-valued points of elliptic curves
Kyushu University
Shinnya Okumura
Outline
ܧ/Է: elliptic curve over Է defined by a Weierstrass equation with integer coefficients.
, κ prime number.
ܧ(ॲ): group of ॲ-valued points of ܧ mod .
(ܧ has good reduction at ).
ܽ, ܾ, ݐ : positive integers with GCD(ܽ, ܾ) = 1.
1. Primality of |ܧ(ॲ)|/ݐ when varies satisfying ؠ ܽ ݉݀ ܾ .
2. Divisibility of |ܧ(ॲ)| when varies satisfying ؠ ܽ ݉݀ ܾ .
3. Explanation of our conjecture about primality of
|ܧ(ॲ)|/ݐ .
4. Three examples.
Primality of | ܧ (ॲ
) |/ ݐ
ܲ ؔ { ݎ݅݉݁}
ܵா ؔ א ܲ|ܧ ݄ܽݏ ܾܽ݀ ݎ݁݀ݑܿݐ݅݊ ܽݐ
ߨா ,௧ ݔ ؔ| ݔ| א ܲ ך ܵா ܽ݊݀ ܧ ॲ /ݐ ݅ݏ ݎ݅݉݁ |
Conjecture1 (Zywina)
ߨா ,௧ ݔ ~ܥா,௧ ݔ
(log ݔ)ଶ
as ݔ ՜ λ, where ܥா,௧ is a constant depending on ܧ and ݐ.
ͤZywina generalized it to the number field case.
Zywina tested his conjecture for some elliptic curves and got better results than Koblitz’s conjecture.
We try to refine Zywina’s conjecture. More precisely, we consider a next problem.
Are there any conditions of that ܧ ॲ /ݐ is
㻌 㻌 㻌 㻌 㻌
apt to become prime ?We find that the probability that ܧ ॲ /ݐ is prime varies with a if we impose the congruence condition
ؠ ܽ ݉݀ ܾ by testing for some elliptic curves.
So we consider the conditional probability
ܲ ܧ ॲ /ݐ is prime ؠ ܽ(݉݀ ܾ)
Our conjecture about primality of ܧ ॲ /ݐ ȭ,:= א ܲ ؠ ܽ(݉݀ ܾ)
ߨ, ݔ ؔ | ݔ| א ȭ, |
ߨா, ௧, , ݔ ؔ
| ݔ א ȭ, ך ܵா ܽ݊݀ ܧ ॲ /ݐ ݅ݏ ݎ݅݉݁ |
Conjecture2
ܲா ݐ, ܽ, ܾ, ݔ ؔ గಶ, , ೌ,್ ௫
గೌ,್ ௫ ܥா,௧ܥா,௧,, ଵ
୪୭ ௫
as ݔ ՜ λ, where ܥா,௧ is the constant which occurs in Zywina’s conjecture, and Cா,௧,, is a constant
depending on ܧ, ݐ, ܽ and ܾ.
Remark
Zywina used the following integral to calculate the value of ߨா ,௧ ݔ .
ߨா ,௧ ݔ ܥா,௧ ୪୭(௨ାଵ)ି୪୭ ௧ଵ ௫
௧ାଵ
ௗ௨
୪୭ ௨ as ݔ ՜ λ. Zywina’s heuristics suggest that this will be a better approximation of ߨா ,௧ ݔ than ܥா,௧ ௫
(୪୭ ௫)మ .
Thus we use the following expression to calculate the expected value of ܲா ݐ, ܽ, ܾ, ݔ .
ܲா ݐ, ܽ, ܾ, ݔ ܥா,௧ܥா,௧,, 1
log(ݑ + 1) െ log ݐ
௫ ௧ାଵ
log݀ݑݑ ߨ(ݔ)
as ݔ ՜ λ, where ߨ ݔ ؔ | ݔ א ܲ |.
Divisibility of ܧ ॲ
In addition we also consider the divisibility of ܧ ॲ
. More precisely, we consider whether or not for a given ܧ , there is a triple (ܽ, ܾ , κ) א Ժ
ଷwit GCD ܽ, ܾ = 1 such that ܧ ॲ
is
divisible by some prime κ if ؠ ܽ (݉݀ ܾ) .
About this, we prove the next theorem.
Theorem3
ȟ
ா: discriminant of ܧ . ݂
ா conductor of Է ȟ
ா.
Suppose that ܧ is not Է -isogenous to an elliptic curve which has non-trivial Է-torsion points.
(i). If ȟ
ாב Է , then there are integers
ܽ
ଵ, ܽ
ଶ, . . . , ܽ
ക(ಶ) మwith ܽ
ء ܽ
݉݀ ݂
ா(݅ ് ݆) such that ܧ ॲ
is divisible by 2 if א
ȭ
,ಶך ܵ
ாand ץ 2݂
ாfor ݅ = 1,2, …
ఝ ಶଶ
,
where ߮ 䞉 is the Euler function.
(ii). If ȟா א Է and ܨா is the conductor of Է ܧ 2 , then there are integers ܽଵ, ܽଶ, . . . , ܽക(ಷಶ)
య
with
ܽء ܽ ݉݀ ܨா (݅ ് ݆) such that ܧ ॲ is divisible by 2 if א ȭ,ிಶ ך ܵா and ץ 2ܨா for
݅ = 1,2, … ఝ ிଷಶ , where the field Է ܧ 2 is obtained from Է by adjoining the coordinates of all points of ܧ 2 .
Explanation of our conjecture
If ܧ ॲ
/ݐ behaves like a random integer
when varies, by the prime number theorem ܧ ॲ
/ݐ is prime with probability
ଵ
୪୭ ா ॲ /௧
=
୪୭ ா ॲଵ ି୪୭ ௧
ൎ
୪୭ ାଵ ି୪୭ ௧ଵ. But ܧ ॲ
/ݐ do not behave like a random
integer (e.g. Sato-Tate conjecture). So the above probability should be corrected as following :
ܥ ୪୭ ାଵ ି୪୭ ௧ଵ .
The constant ܥ is called a correction factor.
We should define ܥ as if it shows how often
E( ) /ݐ becomes prime than a random integer, when varies.
Then ߨா, ௧ ݔ can be expected as follows:
ܥ 1
log + 1 െ log ݐ
בௌಶ ௧ஸஸ௫
ܥ න 1
log ݑ + 1 െ log ݐ
௫ ௧ାଵ
݀ݑ log ݑ ܥ (୪୭ ௫)௫ మ (ݔ ՜ λ).
We should define ܥா, ௧, , as if it shows how often E( ) /ݐ becomes prime than the unconditional case, when varies with ؠ ܽ mod ܾ .
ߨா, ௧, , ݔ can be expected as follows:
1
߮ ܾ ܥா, ௧, , ܥ 1
log + 1 െ log ݐ
בௌಶ ௧ஸஸ௫
1
߮ ܾ ܥா, ௧, , ܥ න 1
log ݑ + 1 െ log ݐ
௫ ௧ାଵ
݀ݑ log ݑ ఝ ଵ ܥா, ௧, , ܥ (୪୭ ௫)௫ మ (ݔ ՜ λ).
1. Zywina’s heuristics
First, we recall some facts about a Galois
representation attached to torsion subgroups of ܧ. ߩ : ܩԷ ื Aut(ܧ ݉ ) ؆ ܩܮଶ(Ժ/݉Ժ)=: ܩ
ܩ ݉ ؔ Imߩ ؆ Gal(Է(ܧ ݉ )/Է)
Frob Frobenius conjugacy class at .
Then, if ץ ݉, ݅ݏ ݑ݊ݎ݂ܽ݉݅݅݁݀ ݅݊ Է ܧ ݉ and
|E( )| ؠ det(ܫ െ ߩ Frob )
=det ߩ Frob + 1 െ T୰(ߩ Frob ) (mod ݉)
Moreover, we recall the Chebotarev density theorem.
Theorem4(Chebotarev)
Let ܮ/ܭ be a (finite) Galois extension of number fields and let ࣝ كGal(ܮ/ܭ) be any subset which is stable by Gal(ܮ/ܭ)-conjugation. Let ܲ be the set of prime ideals of ࣩ where ࣩ is the ring of integers of ܭ, and let ܲ(ݔ) be the set of prime ideals ॣ of ࣩ such that
ܰ(ॣ) ݔ. For each ॣ א ܲ which is unramified in ܮ, let Frobॣ ك Gal(ܮ/ܭ) denote the conjugacy class of the Frobenius element attached to any prime ो of ܮ lying over ॣ. Then
௫՜ஶlim
| ॣ א ܲ ݔ ॣ ݅ݏ ݑ݊ݎ݂ܽ݉݅݅݁݀ ݅݊ ܮ ܽ݊݀Frobॣ ك ࣝ |
|ܲ(ݔ)|
=|Gal|ࣝ|(/)| .
Zywina define a next set
߰௧ ݉ ؔ ܣ א ܩ ݉ | det (ܫ െ ܣ) א ݐ ή (Ժ/݉Ժ) × .
The Chebotarev density theorem implies that when ݉ is divisible by ݐ ς κκ|௧ ,
ߜா,௧ ݉ ؔ |߰௧ ݉ |
|ܩ ݉ |
is the probability that for a random א ܲ ך ܵா
,
E( ) /ݐ is an integer and invertible mod ݉.
On the other hand, the probability that a random natural number is invertible mod ݉ is ςκ|(1 െ ଵκ). He expected
ߜா,௧ ݉ ς (1 െ 1
κ)
κ|
1
log + 1 െ log ݐ to be a better approximation for
ܲ ܧ ॲ /ݐ is prime ݎ݅݉݁ and defined ܥா, ௧ as follows:
ܥா, ௧ ؔ lim
௭՜ஶ
ߜா,௧ ݐ ςκஸ௭ κ ς (1 െ 1
κ)
κஸ௭
2. Our heuristics We define
߰௧, , ݉ ؔ
ܣ א ߰௧ ݉ | det ܣ mod κ = ܽ for all κ צ gcd(݉, ܾ) . ܩ,௧ ݉ ؔ ܣ א ߰௧ ݉ det ܣ = ܽ .
ܾ = ςκ| κ(κ).
㻌
(decomposition into prime factors)݉ ݖ ؔ ςκஸ௭ κ(κ) (if κ ץ ܾ, ݎ κ = 1).
If ݖ > ݐ, ܾ, |ట, ೌ, ್|ீ ௧(௭) |௧(௭) | is the probability that for a random א ܲ ך ܵா
,
E( ) /ݐ is an integer, relatively prime to ݉ ݖ and ؠ ܽ mod ܾ .We define ܥா, ௧, , as follows:
ܥா, ௧, , ؔ lim
௭՜ஶ
߮(ܾ)|߰௧, , ݐ݉(ݖ) |
|ܩ ݐ݉ ݖ |
ߜா,௧ ݐ݉(ݖ) .
ͤIf ܧ has complex multiplication, we define ܥா, ௧, , by a different way, because it is convenient for
calculation of it.
Calculation of ܥ
ா, ௧, , (non-CM case)
Assume that ܧ does not have complex multiplication.
Theorem5(Serre)
There is a constant ܯ ؔ ܯா depending on ܧ such that if ݉ and ݊ are positive integers with
gcd ݊, ݉ܯ = 1, then
ܩ ݉݊ ؆ ܩ ݉ × Aut ܧ ݊ . From now on we assume gcd ܾ, ܯݐ = 1.
From this theorem, we may identify ߰௧, , ݐ݉(ݖ) with the direct product set
ෑ ܩ, ௧ κ κ
κ|,
× ෑ ߰௧ κ
κץெ௧
× ߰௧ ݐ ෑ κ
κ|ெ௧
,
where κ runs over all κ|݉(ݖ). It is easy to see that If κ ץ ܯݐ, then ܩ(κ) = ܩκ and
|ܩ,௧ κ |
|߰௧ κ | = κଷ ିଵ |ܩ,௧ κ |
κସ ିଵ |߰௧ κ | = |ܩ,ଵ κ |
κ ିଵ |߰ଵ κ | . Thus we have
ܥா, ௧, , = lim
௭՜ஶ
߮(ܾ)|߰௧, , ݐ݉(ݖ) |
|߰௧ ݐ݉(ݖ) |
= ෑ(κ κ െ κ( κ ିଵ)) |ܩ,௧ κ κ |
|߰௧ κ κ |
κ|
= ෑ(κ െ 1) |ܩ,ଵ κ |
|߰ଵ κ |
κ|
.
Lemma6
If gcd ܾ, ܯݐ = 1 ,
ܥ
ா, ௧, , = ෑ ܥ
ா, ଵ, ,κκ|
. Theorem7
If gcd ܾ, ܯݐ = 1 , then ܥ
ா, ௧, , can be calculated as follows:
ܥா,௧,, = ቐς κିଵ κమିκିଵ
κయିଶκమିκାଷ
κ| ݂݅ ܽ = 1,
ς κିଶ κమିଵ
κయିଶκమିκାଷ
κ| ݂݅ܽ ് 1.
Calculation of ܥ
ா, ௧, , (CM case)
Theorem8(Serre) ܭ number field.
ܧ/ܭ : elliptic curve defined over ܭ with complex multiplication.
ܴ ؔ Endഥ ܧ .
Assume that all the elements of ܴ are defined over ܭ.
Then there is a constant ܯ ؔ ܯா಼ depending on ܧ such that if ݉ and ݊ are positive integers with
gcd ݊, ݉ܯ = 1, then
ܩ ݉݊ ؆ ܩ ݉ × ܴ
ൗܴ݊ ×.
Assume that ܧ has complex multiplication.
ܴ ؔ EndԷഥ ܧ ؆ Ժ + Ժ݂ߙ.
ܨ ؔ ܴ۪ԺԷ ؆ Է( ܦ) (ܦ : square-free integer) ߙ = ቐ
ܦ if ܦ ؠ 2, 3 mod 4 ,
1 + ܦ
2 if ܦ ؠ 1 mod 4 .
݀ி : ܦ݅ݏܿ(ܨ).
ܲி ؔ ݎ݅݉݁ ݈݅݀݁ܽ ݂ࣩி .
(ࣩி ݄ܶ݁ ݎ݅݊݃ ݂݅݊ݐ݁݃݁ݎ ݂ ܨ)
Not all the elements of ܴ are defined over Է.
֜ We consider ܧ as an elliptic curve over ܨ and denote this curve by ܧி.
ȭி,,ୱ୮୪୧୲ ؔ
א ܲ ך ܵா| ؠ ܽ ݉݀ ܾ , ݏ݈݅ݐݏ ݈ܿ݉݁ݐ݈݁ݕ ݅݊ ܨ .
ȭி,,ୱ୮୪୧୲ ݔ ؔ
ॣ א ܲி | ॣ ת Ժ = , א ȭி,,ୱ୮୪୧୲ ܽ݊݀ ܰி(ॣ) ݔ . ߨா
ಷ, ௧, , ୱ୮୪୧୲ ݔ
ؔ ॣ א ȭி,,ୱ୮୪୧୲ ݔ െ ܵாಷ ||ܧ(ॲॣ)|/ݐ is prime
.
ߨாୱ୮୪୧୲, ி,௧, , ݔ
ؔ | א ȭிୱ୮୪୧୲,,| ݔ ܽ݊݀ |ܧ(ॲ)|/ݐ is prime |.
Similarly, we can define ȭி,,୧୬ୣ୰୲, ȭி,,୧୬ୣ୰୲ ݔ and ߨா୧୬ୣ୰୲, ி,௧, , ݔ . Then we have
ߨா, ௧, , ݔ
=
ߨாୱ୮୪୧୲, ி,௧, , ݔ + ߨா୧୬ୣ୰୲, ி,௧, , ݔ + ܱ(1).ߨா, ி, ௧ୱ୮୪୧୲ , , ݔ = ଵଶ ߨாୱ୮୪୧୲ಷ, ௧, , ݔ . Conjecture9
ߨா
ಷ, ௧, ,
ୱ୮୪୧୲ ݔ ~ܥா
ಷ, ௧, ,
ୱ୮୪୧୲ ߨாಷ, ௧ ݔ ~ܥா
ಷ, ௧, ,
ୱ୮୪୧୲ ܥாಷ, ௧ ݔ
(log ݔ)ଶ ߨா, ி,௧, , ୧୬ୣ୰୲ ݔ ~ܥ௧, , ୧୬ୣ୰୲ ݔ
(log ݔ)ଶ .
ߨா, ௧, , ݔ ~(ଵଶ ܥா
ಷ, ௧, ,
ୱ୮୪୧୲ ܥாಷ, ௧ + ܥ௧, , ୧୬ୣ୰୲) (୪୭ ௫)௫ మ .
(ݔ ื λ) We define ܥா, ௧, , as follows:
ܥா, ௧, , ؔ ߮(ܾ) (ଵଶ ܥா
ಷ, ௧, ,
ୱ୮୪୧୲ ܥாಷ, ௧ + ܥ௧୧୬ୣ୰୲, , )ܥா, ௧ିଵ.
Note that
|ȭி,,ଵ୧୬ୣ୰୲ ݔ |
| ॣ א ܲி | ܰி(ॣ) ݔ | ื 0 (ݔ ื λ).
Thus, if ݖ > ݐ, ܾ
,
then we may consider|߰௧, , ݐ݉(ݖ) |
|ܩ ݐ݉(ݖ) |
to be the probability that for a random ॣ א ܲி ך ܵாಷ, Eி ( ॣ) /ݐ is an integer, relatively prime to ݉ ݖ and
ॣ ת Ժ = , א ȭி,,ୱ୮୪୧୲ . We define
ܥா
ಷ, ௧, ,
ୱ୮୪୧୲ ؔ lim
௭՜ஶ
|߰௧, , ݐ݉(ݖ) |
|ܩ ݐ݉ ݖ |
ߜாಷ,௧ ݐ݉(ݖ) .
Theorem10 Assume that ܯ is divisible by ςκ|ଶௗಷ κ. If κ ץ ܯݐ, then ܥா
ಷ, ௧, ,κೝ κ
ୱ୮୪୧୲ can be calculated as follows:
(i)If κ =1, then we have
ܥாಷ, ௧, ,κೝ κ
ୱ୮୪୧୲ = ቐ
ଵ
κೝ κ షభ(κିଶ) ݂݅ ܽ = 1
κିଷ
κೝ κ షభ(κିଵ)మ ݂݅ ܽ ് 1
(ii)If κ =-1, then we have
ܥா
ಷ, ௧, ,κೝ κ
ୱ୮୪୧୲ = ቐ
κ
κೝ κ షభ(κమିଶ) ݂݅ ܽ = 1
κାଵ
κೝ κ షభ(κమିଶ) ݂݅ ܽ ് 1
Lemma11
ܨ = Է( ܦ), (ܦ squareоfree) . ݂ி conductor of ܨ.
Then there are integers ܽଵ, ܽଶ, . . . , ܽക(ಷ) మ
㻌
(ܽ ء ܽ mod ݂ி (݅ ് ݆)) such that the following holds:
If א ܲ ܽ݊݀ ץ 2݂ிܦ, then is inert in ܨ
ؠ ܽ (mod ݂ி) for some ݅ א 1,2, . . . , ఝ(ಷ)
ଶ .
If is inert in ܨ, then |E( p)|= + 1.
Lemma12
݀ ؔ gcd(ܾ, ݂ி).
ܬ ؔ ݅ א 1,2, . . . , ఝ(ଶಷ) ݀|(ܽ െ ܽ) .
ߨ௧, ఈ, ఉ ݔ ؔ | ݔ | א ܲ, ؠ ߙ mod ߚ , + 1
ݐ ݅ݏ ݎ݅݉݁ |.
Then there are integers ܿ and ߳ such that the following holds:
ߨா୧୬ୣ୰୲, ி,௧, , ݔ =
ߨ௧,,ಷ ݔ
ఝ(ಷ)/ଶ
ୀଵ
+ ܱ 1 ݂݅ ݀ = 1
ߨ௧,ఢ
,ᇲಷ ݔ
א
+ ܱ 1 ݂݅ ݀ > 1, ݂ி = ݂݀ᇱி
Conjecture13
There is a constant ܥఈ, ఉ,௧ depending of ݐ, ߙ and ߚ such that
ߨ௧, ఈ, ఉ ݔ ~ ܥఈ, ఉ,௧
߮(ߚ)
ݔ
(log ݔ)ଶ (ݔ ื λ).
ܥఈ, ఉ,௧is determined as follows:
ܲ + 1
ݐ ݎ݅݉݁ א ܲ, ؠ ߙ mod ߚ =
ܲ ݐ|( + 1) א ܲ, ؠ ߙ mod ߚ ×
ܲ + 1
ݐ ݎ݅݉݁ א ܲ, ؠ ߙ mod ߚ , ݐ|( + 1) .
ܲ + 1
ݐ ݎ݅݉݁ א ܲ, ؠ ߙ mod ߚ , ݐ|( + 1) ൎ ܲ ݊ ݎ݅݉݁ ×
ෑ ܲ κ ץ + 1
ݐ א ܲ, ؠ ߙ mod ߚ , ݐ|( + 1)
ܲ(κ ץ ݊)
κ
, where ܲ ݊ ݎ݅݉݁ is the probability that a random
integer is prime, and ܲ κ ץ ݊ is the probability that a random integer is not divisible by κ.
We define
ܥఈ, ఉ,௧ ؔ ܲ ݐ|( + 1) א ܲ, ؠ ߙ mod ߚ ×
ෑ ܲ κ ץ + 1
ݐ א ܲ, ؠ ߙ mod ߚ , ݐ|( + 1)
ܲ(κ ץ ݊)
κ
.
Theorem14 Assume that gcd ܾ, ݂ி = 1. (i)There are integers ܿ such that
ܥ௧, , ୧୬ୣ୰୲ = ఝ(ଵ
ಷ) σ ܥ,ಷ,௧
ക(ಷ)
ୀଵమ .
(ii-i)If gcd ܾ݂ி, ݐ = 1, there are integers ߙ for
݅ א 1,2, . . . , ఝ(ಷ)
ଶ such that the following holds:
If κ א ܲ such that κ|ܾ݂ி and ݐκ|(ߙ + 1), then ܥ,ಷ,௧= 0. Otherwise
ܥ,ಷ,௧
=
1
߮ ݐ ෑ 1 െ 1
κ െ 1 ଶ ෑ 1 + 1 κ െ 1
κ|ಷ κץಷ௧
݂݅ ݐ| ߙ + 1 , 1
߮ ݐ ෑ 1 െ 1
κ െ 1 ଶ ෑ 1 + 1 κ െ 1
κ|ಷ௧ κץಷ௧
݂݅ ݐ ץ ߙ + 1 .
(ii-ii)If gcd ܾ݂ி, ݐ = ݁ > 1and ܾ݂ி = ݂݁ᇱ, there are integers ߚ for ݅ א 1,2, . . . , ఝ(ଶಷ) such that the
following holds:
If ݁ ץ ܿ + 1 or κ א ܲ such that κ|݂ᇱ and ݐκ|(ߚ+1), thenܥ,ಷ,௧ = 0. Otherwise if ݐ = ݐᇱ݁, gcd(݁, ݐᇱ) = ݄ and ݐᇱ = ݐᇱᇱ݄, then we have
ܥ,ಷ,௧
=
1
߮ ݐᇱᇱ ݄ ෑ 1 െ 1
κ െ 1 ଶ ෑ 1 + 1 κ െ 1
κ|ᇲ κץಷ௧ᇲ
݂݅ ݐ| ߚ+1 , 1
߮ ݐᇱᇱ ݄ ෑ 1 െ 1
κ െ 1 ଶ ෑ 1 + 1 κ െ 1
κ|ಷ௧ᇲ κץಷ௧ᇲ
݂݅ ݐ ץ ߚ+1 .
Example1 (Non-CM case)
ܧ ݕଶ = ݔଷ െ ݔ + 2, ܯ = 6.
ܥா, ଵ =0.5612957424882619712979385….
Moreover
ܥா, ଵ,,ଵହ =
152
73 ݂݅ ܽ = 1 144
73 ݂݅ ܽ = 4, 7,13, 0 ݂݅ ܽ = 2,8,11,14.
ܽ,ܾ = (1, 5)
ܽ,ܾ = (1, 7)
ܽ,ܾ = (0, 1)
ܽ,ܾ = 2, 5 , 3,5 , (4,5)
ܽ,ܾ = 2, 7 , 3,7 , 4,7 , 5,7
ܽ,ܾ = (0, 1)
ܽ,ܾ = (1, 15)
ܽ,ܾ = 4, 15 , 7,15 , (13,15)
Example2 (Non-CM case)
ܧ ݕଶ + ݕ = ݔଷ െ ݔଶ െ ͳͲݔ െ 20, ܯ = 2 ή 5 ή 11.
ܥா, ହ, , ଷ = ൞ 10
9 , 8 9 .
If ܾ = ݐ = 5, then gcd(ܾ, ܯݐ) = 5 ് 1. However, we can calculate ܥா, ହ ,, ହ, because ܩ(2 ή 5ଶ ή 11) is studied by Lang and Trotter, and so we know the
structure of ܩ 2 ή 5ଶ ή 11 . ܥா, ହ, , ହ = ቐ
0 ݂݅ ܽ = 1, 4
3 ݂݅ ܽ ് 1.
ܽ,ܾ = (0, 1)
ܽ,ܾ = (1, 3)
ܽ,ܾ = (2, 3)
ܽ,ܾ = 2 5 , 3,5 , (4,5)
Example2 (CM case)
ܧ ݕଶ = ݔଷ െ ݔ , ܯ = 2, ܴ = Ժ ݅ , ݂ி = 4, ݂ = 1.
ܥா, ଼ =
ܥாԷ(), ଼ = 1.067350894. . .
ܥா, ଼, , ଷ =
ܥா, ଼, , ହ =
ܥா, ଼, , =
ܽ,ܾ = (1, 5)
ܽ,ܾ = (4, 5)
ܽ,ܾ = (1, 3)
ܽ,ܾ = (2, 3)
ܽ,ܾ = 2, 5 , (3,5)
ܽ,ܾ = (0, 1)
ܽ,ܾ = (6, 7)
ܽ,ܾ = (1, 7)
ܽ,ܾ = (0, 1)
ܽ,ܾ = 2, 7 , 3,7 , 4,7 , 5,7 , (6,7)
