sage:E = EllipticCurve([0, -1, 0, -72, -216])
gp:E = ellinit([0, -1, 0, -72, -216])
magma:E := EllipticCurve([0, -1, 0, -72, -216]);
oscar:E = elliptic_curve([0, -1, 0, -72, -216])
sage:E.short_weierstrass_model()
magma:WeierstrassModel(E);
oscar:short_weierstrass_model(E)
trivial
magma:MordellWeilGroup(E);
Invariants
Conductor: |
N |
= |
4704 | = | 25⋅3⋅72 |
sage:E.conductor().factor()
gp:ellglobalred(E)[1]
magma:Conductor(E);
oscar:conductor(E)
|
Discriminant: |
Δ |
= |
−677376 | = | −1⋅29⋅33⋅72 |
sage:E.discriminant().factor()
gp:E.disc
magma:Discriminant(E);
oscar:discriminant(E)
|
j-invariant: |
j |
= |
−271668296 | = | −1⋅23⋅3−3⋅7⋅313 |
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 | ≈ | −0.080772029862700235109813958423 |
gp:ellheight(E)
magma:FaltingsHeight(E);
oscar:faltings_height(E)
|
Stable Faltings height: |
hstable | ≈ | −0.92495077345854476802363017342 |
magma:StableFaltingsHeight(E);
oscar:stable_faltings_height(E)
|
abc quality: |
Q | ≈ | 0.9223157301458332 |
|
Szpiro ratio: |
σm | ≈ | 2.895524208532298 |
|
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 | ≈ | 0.81774419962573641479678708710 |
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 | = | 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) | ≈ | 0.81774419962573641479678708710 |
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);
|
0.817744200≈L(E,1)=#E(Q)tor2#Ш(E/Q)⋅ΩE⋅Reg(E/Q)⋅∏pcp≈121⋅0.817744⋅1.000000⋅1≈0.817744200
sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve([0, -1, 0, -72, -216]); 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, -72, -216]); 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
4704.2.a.b
q−q3−3q5+q9−q11+4q13+3q15−4q17+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.
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 = [[7, 2, 0, 1], [1, 2, 0, 1], [17, 2, 17, 3], [1, 0, 2, 1], [23, 2, 22, 3], [13, 2, 13, 3], [1, 1, 23, 0]]
GL(2,Integers(24)).subgroup(gens)
magma:Gens := [[7, 2, 0, 1], [1, 2, 0, 1], [17, 2, 17, 3], [1, 0, 2, 1], [23, 2, 22, 3], [13, 2, 13, 3], [1, 1, 23, 0]];
sub<GL(2,Integers(24))|Gens>;
The image H:=ρE(Gal(Q/Q)) of the adelic Galois representation has
label 24.2.0.b.1,
level 24=23⋅3, index 2, genus 0, and generators
(7021),(1021),(171723),(1201),(232223),(131323),(12310).
The torsion field K:=Q(E[24]) is a degree-36864 Galois extension of Q with Gal(K/Q) isomorphic to the projection of H to GL2(Z/24Z).
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 |
4 |
147=3⋅72 |
3 |
nonsplit multiplicative |
4 |
1568=25⋅72 |
7 |
additive |
14 |
96=25⋅3 |
gp:ellisomat(E)
This curve has no rational isogenies. Its isogeny class 4704g
consists of this curve only.
This elliptic curve is its own minimal quadratic twist.
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:
[K:Q] |
K |
E(K)tors |
Base change curve |
3 |
3.1.1176.1 |
Z/2Z |
not in database
|
6 |
6.0.33191424.2 |
Z/2Z⊕Z/2Z |
not in database
|
8 |
8.2.16862305517568.10 |
Z/3Z |
not in database
|
12 |
deg 12 |
Z/4Z |
not in database
|
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.