y2+xy+y=x3+x2−6910x−232261
|
(homogenize, simplify) |
y2z+xyz+yz2=x3+x2z−6910xz2−232261z3
|
(dehomogenize, simplify) |
y2=x3−8955387x−10702030122
|
(homogenize, minimize) |
sage:E = EllipticCurve([1, 1, 1, -6910, -232261])
gp:E = ellinit([1, 1, 1, -6910, -232261])
magma:E := EllipticCurve([1, 1, 1, -6910, -232261]);
oscar:E = elliptic_curve([1, 1, 1, -6910, -232261])
sage:E.short_weierstrass_model()
magma:WeierstrassModel(E);
oscar:short_weierstrass_model(E)
trivial
magma:MordellWeilGroup(E);
Invariants
Conductor: |
N |
= |
294 | = | 2⋅3⋅72 |
sage:E.conductor().factor()
gp:ellglobalred(E)[1]
magma:Conductor(E);
oscar:conductor(E)
|
Discriminant: |
Δ |
= |
−1613775332736 | = | −1⋅27⋅37⋅78 |
sage:E.discriminant().factor()
gp:E.disc
magma:Discriminant(E);
oscar:discriminant(E)
|
j-invariant: |
j |
= |
−2799366329617441 | = | −1⋅2−7⋅3−7⋅7⋅9673 |
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 | ≈ | 1.1074316380143106234153900780 |
gp:ellheight(E)
magma:FaltingsHeight(E);
oscar:faltings_height(E)
|
Stable Faltings height: |
hstable | ≈ | −0.18984179468923157998817841763 |
magma:StableFaltingsHeight(E);
oscar:stable_faltings_height(E)
|
abc quality: |
Q | ≈ | 1.032335220248665 |
|
Szpiro ratio: |
σm | ≈ | 6.722775443275639 |
|
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 | ≈ | 0.26114291828811105870441034589 |
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 | = | 7
= 7⋅1⋅1
|
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 | = | 1 |
sage:E.torsion_order()
gp:elltors(E)[1]
magma:Order(TorsionSubgroup(E));
oscar:prod(torsion_structure(E)[1])
|
Special value: |
L(E,1) | ≈ | 1.8280004280167774109308724213 |
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);
|
1.828000428≈L(E,1)=#E(Q)tor2#Ш(E/Q)⋅ΩE⋅Reg(E/Q)⋅∏pcp≈121⋅0.261143⋅1.000000⋅7≈1.828000428
sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve([1, 1, 1, -6910, -232261]); 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, -6910, -232261]); 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
294.2.a.e
q+q2−q3+q4+q5−q6+q8+q9+q10+5q11−q12−q15+q16−4q17+q18+8q19+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 = [[127, 14, 0, 1], [1, 0, 14, 1], [113, 14, 119, 99], [8, 5, 91, 57], [155, 14, 154, 15], [125, 70, 0, 53], [85, 14, 91, 99], [1, 14, 0, 1]]
GL(2,Integers(168)).subgroup(gens)
magma:Gens := [[127, 14, 0, 1], [1, 0, 14, 1], [113, 14, 119, 99], [8, 5, 91, 57], [155, 14, 154, 15], [125, 70, 0, 53], [85, 14, 91, 99], [1, 14, 0, 1]];
sub<GL(2,Integers(168))|Gens>;
The image H:=ρE(Gal(Q/Q)) of the adelic Galois representation has
level 168=23⋅3⋅7, index 96, genus 2, and generators
(1270141),(11401),(1131191499),(891557),(1551541415),(12507053),(85911499),(10141).
The torsion field K:=Q(E[168]) is a degree-1548288 Galois extension of Q with Gal(K/Q) isomorphic to the projection of H to GL2(Z/168Z).
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 |
2 |
split multiplicative |
4 |
147=3⋅72 |
3 |
nonsplit multiplicative |
4 |
98=2⋅72 |
7 |
additive |
26 |
1 |
gp:ellisomat(E)
This curve has non-trivial cyclic isogenies of degree d for d=
7.
Its isogeny class 294.e
consists of 2 curves linked by isogenies of
degree 7.
The minimal quadratic twist of this elliptic curve is
294.f1, its twist by −7.
The number fields K of degree less than 24 such that
E(K)tors is strictly larger than E(Q)tors
(which is trivial)
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.
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.