Properties

Label 25451.c1
Conductor 25451
Discriminant -788981
j-invariant \( -\frac{17434421857}{788981} \)
CM no
Rank 3
Torsion Structure \(\mathrm{Trivial}\)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, -54, 136]); // or
magma: E := EllipticCurve("25451a1");
sage: E = EllipticCurve([1, 1, 1, -54, 136]) # or
sage: E = EllipticCurve("25451a1")
gp: E = ellinit([1, 1, 1, -54, 136]) \\ or
gp: E = ellinit("25451a1")

\( y^2 + x y + y = x^{3} + x^{2} - 54 x + 136 \)

Mordell-Weil group structure

\(\Z^3\)

Infinite order Mordell-Weil generators and heights

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(\frac{61}{16}, \frac{1}{64}\right) \)\( \left(-4, 19\right) \)\( \left(-\frac{13}{4}, \frac{145}{8}\right) \)
\(\hat{h}(P)\) ≈  2.481020141742.273203870793.60691363716

Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

\( \left(-8, 15\right) \), \( \left(-4, 19\right) \), \( \left(-2, 16\right) \), \( \left(2, 5\right) \), \( \left(4, 0\right) \), \( \left(5, 2\right) \), \( \left(13, 36\right) \), \( \left(29, 140\right) \), \( \left(46, 289\right) \), \( \left(277, 4480\right) \), \( \left(680, 17407\right) \), \( \left(2122, 96717\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 25451 \)  =  \(31 \cdot 821\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(-788981 \)  =  \(-1 \cdot 31^{2} \cdot 821 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( -\frac{17434421857}{788981} \)  =  \(-1 \cdot 31^{-2} \cdot 821^{-1} \cdot 2593^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
Rank: \(3\)
magma: Regulator(E);
sage: E.regulator()
Regulator: \(0.85060994751\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(2.80537676735\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
Tamagawa product: \( 2 \)  = \( 2\cdot1 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
Torsion order: \(1\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 25451.2.a.c

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

\( q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} - 4q^{7} + 3q^{8} - 2q^{9} + 2q^{10} - 3q^{11} + q^{12} - 2q^{13} + 4q^{14} + 2q^{15} - q^{16} - 8q^{17} + 2q^{18} - 8q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 4160
\( \Gamma_0(N) \)-optimal: yes
Manin constant: 1

Special L-value

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

\( L^{(3)}(E,1)/3! \) ≈ \( 4.77256276964 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(31\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(821\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1

Galois representations

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

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

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]

Note: \(p\)-adic regulator data only exists for primes \(p\ge5\) of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 821
Reduction type ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary nonsplit ordinary ordinary ordinary ordinary nonsplit
$\lambda$-invariant(s) 13 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
$\mu$-invariant(s) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Isogenies

This curve has no rational isogenies. Its isogeny class 25451.c 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 3.1.3284.1 \(\Z/2\Z\) Not in database
6 6.0.35416810304.1 \(\Z/2\Z \times \Z/2\Z\) 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.