Properties

Label 12495n1
Conductor 12495
Discriminant -264319707210552871171875
j-invariant \( -\frac{11926249134908509075308544}{2246680441062421875} \)
CM no
Rank 0
Torsion Structure \(\mathrm{Trivial}\)

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Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, -233232715, -1371284647694]) # or
 
sage: E = EllipticCurve("12495n1")
 
gp: E = ellinit([0, 1, 1, -233232715, -1371284647694]) \\ or
 
gp: E = ellinit("12495n1")
 
magma: E := EllipticCurve([0, 1, 1, -233232715, -1371284647694]); // or
 
magma: E := EllipticCurve("12495n1");
 

\( y^2 + y = x^{3} + x^{2} - 233232715 x - 1371284647694 \)

Mordell-Weil group structure

Trivial

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 
None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 12495 \)  =  \(3 \cdot 5 \cdot 7^{2} \cdot 17\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-264319707210552871171875 \)  =  \(-1 \cdot 3^{10} \cdot 5^{7} \cdot 7^{9} \cdot 17^{5} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{11926249134908509075308544}{2246680441062421875} \)  =  \(-1 \cdot 2^{15} \cdot 3^{-10} \cdot 5^{-7} \cdot 7^{-3} \cdot 17^{-5} \cdot 7139777^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(0\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.019316003869\)
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: \( 140 \)  = \( ( 2 \cdot 5 )\cdot7\cdot2\cdot1 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(1\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 12495.2.a.h

sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 
magma: ModularForm(E);
 

\( q + q^{3} - 2q^{4} + q^{5} + q^{9} + 2q^{11} - 2q^{12} + 5q^{13} + q^{15} + 4q^{16} - q^{17} - 2q^{19} + O(q^{20}) \)

For more coefficients, see the Downloads section to the right.

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 2016000
\( \Gamma_0(N) \)-optimal: yes
Manin constant: 1

Special L-value

sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 

\( L(E,1) \) ≈ \( 2.70424054165 \)

Local data

This elliptic curve is not semistable.

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(3\) \(10\) \( I_{10} \) Split multiplicative -1 1 10 10
\(5\) \(7\) \( I_{7} \) Split multiplicative -1 1 7 7
\(7\) \(2\) \( I_3^{*} \) Additive -1 2 9 3
\(17\) \(1\) \( I_{5} \) Non-split multiplicative 1 1 5 5

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) .

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
 

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ss split split add ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary
$\lambda$-invariant(s) 3,4 1 3 - 0 0 0 0 0 0 0 0 0 0,0 2
$\mu$-invariant(s) 0,0 0 0 - 0 0 0 0 0 0 0 0 0 0,0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has no rational isogenies. Its isogeny class 12495n consists of this curve only.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.2380.1 \(\Z/2\Z\) Not in database
6 6.0.3370318000.1 \(\Z/2\Z \times \Z/2\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.