sage: E = EllipticCurve([0, 0, 0, -875, -3750])
gp: E = ellinit([0, 0, 0, -875, -3750])
magma: E := EllipticCurve([0, 0, 0, -875, -3750]);
oscar: E = elliptic_curve([0, 0, 0, -875, -3750])
sage: E.short_weierstrass_model()
magma: WeierstrassModel(E);
oscar: short_weierstrass_model(E)
Z \Z Z
magma: MordellWeilGroup(E);
P P P h ^ ( P ) \hat{h}(P) h ^ ( P ) Order
( − 25 , 50 ) (-25, 50) ( − 2 5 , 5 0 ) 0.73981287801200195955783303508 0.73981287801200195955783303508 0 . 7 3 9 8 1 2 8 7 8 0 1 2 0 0 1 9 5 9 5 5 7 8 3 3 0 3 5 0 8 ∞ \infty ∞
( − 25 , ± 50 ) (-25,\pm 50) ( − 2 5 , ± 5 0 ) , ( 150 , ± 1800 ) (150,\pm 1800) ( 1 5 0 , ± 1 8 0 0 )
sage: E.integral_points()
magma: IntegralPoints(E);
Invariants
Conductor :
N N N
=
9200 9200 9 2 0 0 = 2 4 ⋅ 5 2 ⋅ 23 2^{4} \cdot 5^{2} \cdot 23 2 4 ⋅ 5 2 ⋅ 2 3
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
Discriminant :
Δ \Delta Δ
=
36800000000 36800000000 3 6 8 0 0 0 0 0 0 0 0 = 2 12 ⋅ 5 8 ⋅ 23 2^{12} \cdot 5^{8} \cdot 23 2 1 2 ⋅ 5 8 ⋅ 2 3
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
j-invariant :
j j j
=
46305 23 \frac{46305}{23} 2 3 4 6 3 0 5 = 3 3 ⋅ 5 ⋅ 7 3 ⋅ 2 3 − 1 3^{3} \cdot 5 \cdot 7^{3} \cdot 23^{-1} 3 3 ⋅ 5 ⋅ 7 3 ⋅ 2 3 − 1
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
Endomorphism ring :
E n d ( E ) \mathrm{End}(E) E n d ( E ) = Z \Z Z
Geometric endomorphism ring :
E n d ( E Q ‾ ) \mathrm{End}(E_{\overline{\Q}}) E n d ( E Q )
=
Z \Z Z
(no potential complex multiplication )
sage: E.has_cm()
magma: HasComplexMultiplication(E);
Sato-Tate group :
S T ( E ) \mathrm{ST}(E) S T ( E ) = S U ( 2 ) \mathrm{SU}(2) S U ( 2 )
Faltings height :
h F a l t i n g s h_{\mathrm{Faltings}} h F a l t i n g s ≈ 0.72036987451805221494271715871 0.72036987451805221494271715871 0 . 7 2 0 3 6 9 8 7 4 5 1 8 0 5 2 2 1 4 9 4 2 7 1 7 1 5 8 7 1
gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
Stable Faltings height :
h s t a b l e h_{\mathrm{stable}} h s t a b l e ≈ − 1.0457359143312933442083545182 -1.0457359143312933442083545182 − 1 . 0 4 5 7 3 5 9 1 4 3 3 1 2 9 3 3 4 4 2 0 8 3 5 4 5 1 8 2
magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
a b c abc a b c quality :
Q Q Q ≈ 0.8779344571427145 0.8779344571427145 0 . 8 7 7 9 3 4 4 5 7 1 4 2 7 1 4 5
Szpiro ratio :
σ m \sigma_{m} σ m ≈ 3.499114603336856 3.499114603336856 3 . 4 9 9 1 1 4 6 0 3 3 3 6 8 5 6
Analytic rank :
r a n r_{\mathrm{an}} r a n = 1 1 1
sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
Mordell-Weil rank :
r r r = 1 1 1
sage: E.rank()
gp: [lower,upper] = ellrank(E)
magma: Rank(E);
Regulator :
R e g ( E / Q ) \mathrm{Reg}(E/\Q) R e g ( E / Q ) ≈ 0.73981287801200195955783303508 0.73981287801200195955783303508 0 . 7 3 9 8 1 2 8 7 8 0 1 2 0 0 1 9 5 9 5 5 7 8 3 3 0 3 5 0 8
sage: E.regulator()
gp: G = E.gen \\ if available
matdet(ellheightmatrix(E,G))
magma: Regulator(E);
Real period :
Ω \Omega Ω ≈ 0.92392949941871519792253190380 0.92392949941871519792253190380 0 . 9 2 3 9 2 9 4 9 9 4 1 8 7 1 5 1 9 7 9 2 2 5 3 1 9 0 3 8 0
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 :
∏ p c p \prod_{p}c_p ∏ p c p = 6 6 6
= 2 ⋅ 3 ⋅ 1 2\cdot3\cdot1 2 ⋅ 3 ⋅ 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 ) t o r \#E(\Q)_{\mathrm{tor}} # E ( Q ) t o r = 1 1 1
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
Special value :
L ′ ( E , 1 ) L'(E,1) L ′ ( E , 1 ) ≈ 4.1012096522708878927423217358 4.1012096522708878927423217358 4 . 1 0 1 2 0 9 6 5 2 2 7 0 8 8 7 8 9 2 7 4 2 3 2 1 7 3 5 8
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 :
Шa n {}_{\mathrm{an}} a n
≈
1 1 1
(rounded )
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
4.101209652 ≈ L ′ ( E , 1 ) = # Ш ( E / Q ) ⋅ Ω E ⋅ R e g ( E / Q ) ⋅ ∏ p c p # E ( Q ) t o r 2 ≈ 1 ⋅ 0.923929 ⋅ 0.739813 ⋅ 6 1 2 ≈ 4.101209652 \begin{aligned} 4.101209652 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.923929 \cdot 0.739813 \cdot 6}{1^2} \\ & \approx 4.101209652\end{aligned} 4 . 1 0 1 2 0 9 6 5 2 ≈ L ′ ( E , 1 ) = # E ( Q ) t o r 2 # Ш ( E / Q ) ⋅ Ω E ⋅ R e g ( E / Q ) ⋅ ∏ p c p ≈ 1 2 1 ⋅ 0 . 9 2 3 9 2 9 ⋅ 0 . 7 3 9 8 1 3 ⋅ 6 ≈ 4 . 1 0 1 2 0 9 6 5 2
sage: # self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve([0, 0, 0, -875, -3750]); 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, 0, 0, -875, -3750]); 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
9200.2.a.r
q − q 7 − 3 q 9 + q 11 + q 13 + 5 q 19 + O ( q 20 ) q - q^{7} - 3 q^{9} + q^{11} + q^{13} + 5 q^{19} + O(q^{20}) q − q 7 − 3 q 9 + q 1 1 + q 1 3 + 5 q 1 9 + O ( q 2 0 )
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 p 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 \ell ℓ -adic Galois representation has maximal image
for all primes ℓ \ell 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 = [[5, 2, 5, 3], [1, 2, 0, 1], [1, 0, 2, 1], [47, 2, 47, 3], [91, 2, 90, 3], [1, 1, 91, 0]]
GL(2,Integers(92)).subgroup(gens)
magma: Gens := [[5, 2, 5, 3], [1, 2, 0, 1], [1, 0, 2, 1], [47, 2, 47, 3], [91, 2, 90, 3], [1, 1, 91, 0]];
sub<GL(2,Integers(92))|Gens>;
The image H : = ρ E ( Gal ( Q ‾ / Q ) ) H:=\rho_E(\Gal(\overline{\Q}/\Q)) H : = ρ E ( G a l ( Q / Q ) ) of the adelic Galois representation has
level 92 = 2 2 ⋅ 23 92 = 2^{2} \cdot 23 9 2 = 2 2 ⋅ 2 3 , index 2 2 2 , genus 0 0 0 , and generators
( 5 2 5 3 ) , ( 1 2 0 1 ) , ( 1 0 2 1 ) , ( 47 2 47 3 ) , ( 91 2 90 3 ) , ( 1 1 91 0 ) \left(\begin{array}{rr}
5 & 2 \\
5 & 3
\end{array}\right),\left(\begin{array}{rr}
1 & 2 \\
0 & 1
\end{array}\right),\left(\begin{array}{rr}
1 & 0 \\
2 & 1
\end{array}\right),\left(\begin{array}{rr}
47 & 2 \\
47 & 3
\end{array}\right),\left(\begin{array}{rr}
91 & 2 \\
90 & 3
\end{array}\right),\left(\begin{array}{rr}
1 & 1 \\
91 & 0
\end{array}\right) ( 5 5 2 3 ) , ( 1 0 2 1 ) , ( 1 2 0 1 ) , ( 4 7 4 7 2 3 ) , ( 9 1 9 0 2 3 ) , ( 1 9 1 1 0 ) .
The torsion field K : = Q ( E [ 92 ] ) K:=\Q(E[92]) K : = Q ( E [ 9 2 ] ) is a degree-12824064 12824064 1 2 8 2 4 0 6 4 Galois extension of Q \Q Q with Gal ( K / Q ) \Gal(K/\Q) G a l ( K / Q ) isomorphic to the projection of H H H to GL 2 ( Z / 92 Z ) \GL_2(\Z/92\Z) GL 2 ( Z / 9 2 Z ) .
The table below list all primes ℓ \ell ℓ for which the Serre invariants associated to the mod-ℓ \ell ℓ Galois representation are exceptional.
ℓ \ell ℓ
Reduction type
Serre weight
Serre conductor
2 2 2
additive
2 2 2
575 = 5 2 ⋅ 23 575 = 5^{2} \cdot 23 5 7 5 = 5 2 ⋅ 2 3
5 5 5
additive
14 14 1 4
368 = 2 4 ⋅ 23 368 = 2^{4} \cdot 23 3 6 8 = 2 4 ⋅ 2 3
23 23 2 3
nonsplit multiplicative
24 24 2 4
400 = 2 4 ⋅ 5 2 400 = 2^{4} \cdot 5^{2} 4 0 0 = 2 4 ⋅ 5 2
gp: ellisomat(E)
This curve has no rational isogenies. Its isogeny class 9200be
consists of this curve only.
The minimal quadratic twist of this elliptic curve is
575a1 , its twist by − 20 -20 − 2 0 .
The number fields K K K of degree less than 24 such that
E ( K ) t o r s E(K)_{\rm tors} E ( K ) t o r s is strictly larger than E ( Q ) t o r s E(\Q)_{\rm tors} E ( Q ) t o r s
(which is trivial)
are as follows:
[ K : Q ] [K:\Q] [ K : Q ]
K K K
E ( K ) t o r s E(K)_{\rm tors} E ( K ) t o r s
Base change curve
3 3 3
3.3.2300.1
Z / 2 Z \Z/2\Z Z / 2 Z
not in database
6 6 6
6.6.486680000.1
Z / 2 Z ⊕ Z / 2 Z \Z/2\Z \oplus \Z/2\Z Z / 2 Z ⊕ Z / 2 Z
not in database
8 8 8
deg 8
Z / 3 Z \Z/3\Z Z / 3 Z
not in database
12 12 1 2
deg 12
Z / 4 Z \Z/4\Z Z / 4 Z
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 p p -adic regulators
Note: p p p -adic regulator data only exists for primes p ≥ 5 p\ge 5 p ≥ 5 of good ordinary
reduction.
Choose a prime...
13
17
19
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97