y2=x3+x2+45975x+2728827
|
(homogenize, simplify) |
y2z=x3+x2z+45975xz2+2728827z3
|
(dehomogenize, simplify) |
y2=x3+3723948x+1978143012
|
(homogenize, minimize) |
sage:E = EllipticCurve([0, 1, 0, 45975, 2728827])
gp:E = ellinit([0, 1, 0, 45975, 2728827])
magma:E := EllipticCurve([0, 1, 0, 45975, 2728827]);
oscar:E = elliptic_curve([0, 1, 0, 45975, 2728827])
sage:E.short_weierstrass_model()
magma:WeierstrassModel(E);
oscar:short_weierstrass_model(E)
trivial
magma:MordellWeilGroup(E);
Invariants
Conductor: |
N |
= |
383496 | = | 23⋅3⋅19⋅292 |
sage:E.conductor().factor()
gp:ellglobalred(E)[1]
magma:Conductor(E);
oscar:conductor(E)
|
Discriminant: |
Δ |
= |
−9400073837734656 | = | −1⋅28⋅32⋅193⋅296 |
sage:E.discriminant().factor()
gp:E.disc
magma:Discriminant(E);
oscar:discriminant(E)
|
j-invariant: |
j |
= |
6173170575104 | = | 210⋅3−2⋅19−3⋅413 |
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 | ≈ | 1.7524526437778705239732540454 |
gp:ellheight(E)
magma:FaltingsHeight(E);
oscar:faltings_height(E)
|
Stable Faltings height: |
hstable | ≈ | −0.39329339158866336256320338509 |
magma:StableFaltingsHeight(E);
oscar:stable_faltings_height(E)
|
abc quality: |
Q | ≈ | 0.9505813097944765 |
|
Szpiro ratio: |
σm | ≈ | 3.4083264004454437 |
|
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 | ≈ | 0.26653913719380540141728536579 |
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 | = | 16
= 22⋅2⋅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 | = | 1 |
sage:E.torsion_order()
gp:elltors(E)[1]
magma:Order(TorsionSubgroup(E));
oscar:prod(torsion_structure(E)[1])
|
Special value: |
L(E,1) | ≈ | 4.2646261951008864226765658526 |
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);
|
4.264626195≈L(E,1)=#E(Q)tor2#Ш(E/Q)⋅ΩE⋅Reg(E/Q)⋅∏pcp≈121⋅0.266539⋅1.000000⋅16≈4.264626195
sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve([0, 1, 0, 45975, 2728827]); 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, 45975, 2728827]); 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
383496.2.a.t
q+q3+q5−3q7+q9+5q11−2q13+q15+q17−q19+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.
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 = [[37, 2, 36, 3], [1, 2, 0, 1], [1, 0, 2, 1], [21, 2, 21, 3], [1, 1, 37, 0]]
GL(2,Integers(38)).subgroup(gens)
magma:Gens := [[37, 2, 36, 3], [1, 2, 0, 1], [1, 0, 2, 1], [21, 2, 21, 3], [1, 1, 37, 0]];
sub<GL(2,Integers(38))|Gens>;
The image H:=ρE(Gal(Q/Q)) of the adelic Galois representation has
label 38.2.0.a.1,
level 38=2⋅19, index 2, genus 0, and generators
(373623),(1021),(1201),(212123),(13710).
The torsion field K:=Q(E[38]) is a degree-369360 Galois extension of Q with Gal(K/Q) isomorphic to the projection of H to GL2(Z/38Z).
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 |
15979=19⋅292 |
3 |
split multiplicative |
4 |
6728=23⋅292 |
19 |
nonsplit multiplicative |
20 |
20184=23⋅3⋅292 |
29 |
additive |
422 |
456=23⋅3⋅19 |
gp:ellisomat(E)
This curve has no rational isogenies. Its isogeny class 383496t
consists of this curve only.
The minimal quadratic twist of this elliptic curve is
456d1, its twist by 29.
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 |
3.1.76.1 |
Z/2Z |
not in database
|
6 |
6.0.109744.2 |
Z/2Z⊕Z/2Z |
not in database
|
8 |
deg 8 |
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.
No Iwasawa invariant data is available for this curve.
p-adic regulators
All p-adic regulators are identically 1 since the rank is 0.