y2=x3+x2−280x−2440800
|
(homogenize, simplify) |
y2z=x3+x2z−280xz2−2440800z3
|
(dehomogenize, simplify) |
y2=x3−22707x−1779275106
|
(homogenize, minimize) |
sage:E = EllipticCurve([0, 1, 0, -280, -2440800])
gp:E = ellinit([0, 1, 0, -280, -2440800])
magma:E := EllipticCurve([0, 1, 0, -280, -2440800]);
oscar:E = elliptic_curve([0, 1, 0, -280, -2440800])
sage:E.short_weierstrass_model()
magma:WeierstrassModel(E);
oscar:short_weierstrass_model(E)
Z⊕Z/2Z
magma:MordellWeilGroup(E);
P | h^(P) | Order |
(148,884) | 3.1830732673512651251985437098 | ∞ |
(135,0) | 0 | 2 |
(135,0), (148,±884), (4340,±285940)
sage:E.integral_points()
magma:IntegralPoints(E);
Invariants
Conductor: |
N |
= |
437320 | = | 23⋅5⋅13⋅292 |
sage:E.conductor().factor()
gp:ellglobalred(E)[1]
magma:Conductor(E);
oscar:conductor(E)
|
Discriminant: |
Δ |
= |
−2573443615974400 | = | −1⋅210⋅52⋅132⋅296 |
sage:E.discriminant().factor()
gp:E.disc
magma:Discriminant(E);
oscar:discriminant(E)
|
j-invariant: |
j |
= |
−42254 | = | −1⋅22⋅5−2⋅13−2 |
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 | ≈ | 1.6358856107320030560622405829 |
gp:ellheight(E)
magma:FaltingsHeight(E);
oscar:faltings_height(E)
|
Stable Faltings height: |
hstable | ≈ | −0.62538495472785504871042220116 |
magma:StableFaltingsHeight(E);
oscar:stable_faltings_height(E)
|
abc quality: |
Q | ≈ | 1.0943056888733467 |
|
Szpiro ratio: |
σm | ≈ | 3.305924836404655 |
|
Analytic rank: |
ran | = | 1
|
sage:E.analytic_rank()
gp:ellanalyticrank(E)
magma:AnalyticRank(E);
|
Mordell-Weil rank: |
r | = | 1
|
sage:E.rank()
gp:[lower,upper] = ellrank(E)
magma:Rank(E);
|
Regulator: |
Reg(E/Q) | ≈ | 3.1830732673512651251985437098 |
sage:E.regulator()
gp:G = E.gen \\ if available
matdet(ellheightmatrix(E,G))
magma:Regulator(E);
|
Real period: |
Ω | ≈ | 0.20897299646983250431337716221 |
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 | = | 32
= 2⋅2⋅2⋅22
|
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) | ≈ | 5.3214108692913131377253086628 |
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
(rounded)
|
sage:E.sha().an_numerical()
magma:MordellWeilShaInformation(E);
|
5.321410869≈L′(E,1)=#E(Q)tor2#Ш(E/Q)⋅ΩE⋅Reg(E/Q)⋅∏pcp≈221⋅0.208973⋅3.183073⋅32≈5.321410869
sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve([0, 1, 0, -280, -2440800]); 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([0, 1, 0, -280, -2440800]); 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
437320.2.a.d
q−2q3+q5+q9−2q11+q13−2q15−2q17−2q19+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 4 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 = [[3, 8, 10, 27], [1, 0, 8, 1], [15073, 8, 15072, 9], [1, 8, 0, 1], [7799, 0, 0, 15079], [11311, 2088, 2407, 4177], [7541, 2088, 0, 1], [3017, 1566, 0, 1], [5, 8, 48, 77], [10441, 1566, 0, 1]]
GL(2,Integers(15080)).subgroup(gens)
magma:Gens := [[3, 8, 10, 27], [1, 0, 8, 1], [15073, 8, 15072, 9], [1, 8, 0, 1], [7799, 0, 0, 15079], [11311, 2088, 2407, 4177], [7541, 2088, 0, 1], [3017, 1566, 0, 1], [5, 8, 48, 77], [10441, 1566, 0, 1]];
sub<GL(2,Integers(15080))|Gens>;
The image H:=ρE(Gal(Q/Q)) of the adelic Galois representation has
level 15080=23⋅5⋅13⋅29, index 48, genus 0, and generators
(310827),(1801),(150731507289),(1081),(77990015079),(11311240720884177),(7541020881),(3017015661),(548877),(10441015661).
The torsion field K:=Q(E[15080]) is a degree-274574632550400 Galois extension of Q with Gal(K/Q) isomorphic to the projection of H to GL2(Z/15080Z).
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 |
additive |
2 |
841=292 |
5 |
split multiplicative |
6 |
87464=23⋅13⋅292 |
13 |
split multiplicative |
14 |
33640=23⋅5⋅292 |
29 |
additive |
422 |
520=23⋅5⋅13 |
gp:ellisomat(E)
This curve has non-trivial cyclic isogenies of degree d for d=
2.
Its isogeny class 437320d
consists of 2 curves linked by isogenies of
degree 2.
The minimal quadratic twist of this elliptic curve is
520b2, its twist by 29.
No Iwasawa invariant data is available for this curve.
p-adic regulators
p-adic regulators are not yet computed for curves that are not Γ0-optimal.