y2+xy+y=x3−x2−158x−122668
|
(homogenize, simplify) |
y2z+xyz+yz2=x3−x2z−158xz2−122668z3
|
(dehomogenize, simplify) |
y2=x3−2523x−7853258
|
(homogenize, minimize) |
sage:E = EllipticCurve([1, -1, 1, -158, -122668])
gp:E = ellinit([1, -1, 1, -158, -122668])
magma:E := EllipticCurve([1, -1, 1, -158, -122668]);
oscar:E = elliptic_curve([1, -1, 1, -158, -122668])
sage:E.short_weierstrass_model()
magma:WeierstrassModel(E);
oscar:short_weierstrass_model(E)
Z/2Z
magma:MordellWeilGroup(E);
(51,−26)
sage:E.integral_points()
magma:IntegralPoints(E);
Invariants
Conductor: |
N |
= |
37845 | = | 32⋅5⋅292 |
sage:E.conductor().factor()
gp:ellglobalred(E)[1]
magma:Conductor(E);
oscar:conductor(E)
|
Discriminant: |
Δ |
= |
−6504393015135 | = | −1⋅37⋅5⋅296 |
sage:E.discriminant().factor()
gp:E.disc
magma:Discriminant(E);
oscar:discriminant(E)
|
j-invariant: |
j |
= |
−151 | = | −1⋅3−1⋅5−1 |
sage:E.j_invariant().factor()
gp:E.j
magma:jInvariant(E);
oscar:j_invariant(E)
|
Endomorphism ring: |
End(E) | = | Z |
Geometric endomorphism ring: |
End(EQ) |
= |
Z
(no potential complex multiplication)
|
sage:E.has_cm()
magma:HasComplexMultiplication(E);
|
Sato-Tate group: |
ST(E) | = | SU(2) |
Faltings height: |
hFaltings | ≈ | 1.1375293066845027624284300160 |
gp:ellheight(E)
magma:FaltingsHeight(E);
oscar:faltings_height(E)
|
Stable Faltings height: |
hstable | ≈ | −1.0954247526427890968608286186 |
magma:StableFaltingsHeight(E);
oscar:stable_faltings_height(E)
|
abc quality: |
Q | ≈ | 1.1980768440515948 |
|
Szpiro ratio: |
σm | ≈ | 3.5060588538031356 |
|
Analytic rank: |
ran | = | 0
|
sage:E.analytic_rank()
gp:ellanalyticrank(E)
magma:AnalyticRank(E);
|
Mordell-Weil rank: |
r | = | 0
|
sage:E.rank()
gp:[lower,upper] = ellrank(E)
magma:Rank(E);
|
Regulator: |
Reg(E/Q) | = | 1 |
sage:E.regulator()
gp:G = E.gen \\ if available
matdet(ellheightmatrix(E,G))
magma:Regulator(E);
|
Real period: |
Ω | ≈ | 0.34227025899696415732802111833 |
sage:E.period_lattice().omega()
gp:if(E.disc>0,2,1)*E.omega[1]
magma:(Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
|
Tamagawa product: |
∏pcp | = | 8
= 22⋅1⋅2
|
sage:E.tamagawa_numbers()
gp:gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma:TamagawaNumbers(E);
oscar:tamagawa_numbers(E)
|
Torsion order: |
#E(Q)tor | = | 2 |
sage:E.torsion_order()
gp:elltors(E)[1]
magma:Order(TorsionSubgroup(E));
oscar:prod(torsion_structure(E)[1])
|
Special value: |
L(E,1) | ≈ | 0.68454051799392831465604223665 |
sage:r = E.rank();
E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp:[r,L1r] = ellanalyticrank(E); L1r/r!
magma:Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
Analytic order of Ш: |
Шan |
= |
1
(exact)
|
sage:E.sha().an_numerical()
magma:MordellWeilShaInformation(E);
|
0.684540518≈L(E,1)=#E(Q)tor2#Ш(E/Q)⋅ΩE⋅Reg(E/Q)⋅∏pcp≈221⋅0.342270⋅1.000000⋅8≈0.684540518
sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve([1, -1, 1, -158, -122668]); r = E.rank(); ar = E.analytic_rank(); assert r == ar;
Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();
omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();
assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
magma:/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */
E := EllipticCurve([1, -1, 1, -158, -122668]); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;
sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);
reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);
assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
Modular form
37845.2.a.d
q−q2−q4−q5+3q8+q10−4q11−2q13−q16+2q17−4q19+O(q20)
sage:E.q_eigenform(20)
gp:\\ actual modular form, use for small N
[mf,F] = mffromell(E)
Ser(mfcoefs(mf,20),q)
\\ or just the series
Ser(ellan(E,20),q)*q
magma:ModularForm(E);
For more coefficients, see the Downloads section to the right.
This elliptic curve is not semistable.
There
are 3 primes p
of bad reduction:
sage:E.local_data()
gp:ellglobalred(E)[5]
magma:[LocalInformation(E,p) : p in BadPrimes(E)];
oscar:[(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
The ℓ-adic Galois representation has maximal image
for all primes ℓ except those listed in the table below.
sage:rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
magma:[GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage:gens = [[1, 0, 32, 1], [1, 32, 0, 1], [5, 28, 68, 381], [6903, 8642, 11194, 5743], [1654, 12963, 4205, 12268], [4319, 0, 0, 13919], [23, 18, 11358, 11915], [608, 13891, 10933, 318], [13889, 32, 13888, 33], [1, 12992, 13050, 4351]]
GL(2,Integers(13920)).subgroup(gens)
magma:Gens := [[1, 0, 32, 1], [1, 32, 0, 1], [5, 28, 68, 381], [6903, 8642, 11194, 5743], [1654, 12963, 4205, 12268], [4319, 0, 0, 13919], [23, 18, 11358, 11915], [608, 13891, 10933, 318], [13889, 32, 13888, 33], [1, 12992, 13050, 4351]];
sub<GL(2,Integers(13920))|Gens>;
The image H:=ρE(Gal(Q/Q)) of the adelic Galois representation has
level 13920=25⋅3⋅5⋅29, index 768, genus 13, and generators
(13201),(10321),(56828381),(69031119486425743),(165442051296312268),(43190013919),(23113581811915),(6081093313891318),(13889138883233),(113050129924351).
The torsion field K:=Q(E[13920]) is a degree-8046143078400 Galois extension of Q with Gal(K/Q) isomorphic to the projection of H to GL2(Z/13920Z).
The table below list all primes ℓ for which the Serre invariants associated to the mod-ℓ Galois representation are exceptional.
ℓ |
Reduction type |
Serre weight |
Serre conductor |
3 |
additive |
8 |
4205=5⋅292 |
5 |
nonsplit multiplicative |
6 |
7569=32⋅292 |
29 |
additive |
422 |
45=32⋅5 |
gp:ellisomat(E)
This curve has non-trivial cyclic isogenies of degree d for d=
2, 4, 8 and 16.
Its isogeny class 37845d
consists of 8 curves linked by isogenies of
degrees dividing 16.
The minimal quadratic twist of this elliptic curve is
15a8, its twist by −87.
The number fields K of degree less than 24 such that
E(K)tors is strictly larger than E(Q)tors
≅Z/2Z
are as follows:
We only show fields where the torsion growth is primitive.
For fields not in the database, click on the degree shown to reveal the defining polynomial.
All Iwasawa λ and μ-invariants for primes p≥3 of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
p-adic regulators
All p-adic regulators are identically 1 since the rank is 0.