Properties

Label 31737a1
Conductor 31737
Discriminant -95211
j-invariant \( -\frac{4475809792}{95211} \)
CM no
Rank 3
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([0, -1, 1, -34, 90]); // or
magma: E := EllipticCurve("31737a1");
sage: E = EllipticCurve([0, -1, 1, -34, 90]) # or
sage: E = EllipticCurve("31737a1")
gp: E = ellinit([0, -1, 1, -34, 90]) \\ or
gp: E = ellinit("31737a1")

\( y^2 + y = x^{3} - x^{2} - 34 x + 90 \)

Mordell-Weil group structure

\(\Z^3\)

Infinite order Mordell-Weil generators and heights

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(-6, 6\right) \)\( \left(-5, 10\right) \)\( \left(4, 1\right) \)
\(\hat{h}(P)\) ≈  2.291994814311.675885217290.517783667116

Integral points

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

\( \left(-6, 6\right) \), \( \left(-5, 10\right) \), \( \left(-3, 12\right) \), \( \left(0, 9\right) \), \( \left(1, 7\right) \), \( \left(3, 2\right) \), \( \left(4, 1\right) \), \( \left(5, 4\right) \), \( \left(10, 25\right) \), \( \left(14, 46\right) \), \( \left(22, 97\right) \), \( \left(28, 142\right) \), \( \left(289, 4903\right) \), \( \left(301, 5212\right) \), \( \left(1551, 61062\right) \), \( \left(710060, 598332049\right) \)

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 31737 \)  =  \(3 \cdot 71 \cdot 149\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(-95211 \)  =  \(-1 \cdot 3^{2} \cdot 71 \cdot 149 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( -\frac{4475809792}{95211} \)  =  \(-1 \cdot 2^{12} \cdot 3^{-2} \cdot 71^{-1} \cdot 103^{3} \cdot 149^{-1}\)
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: \(0.624153583024\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(3.37760876498\)
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: \( 2 \)  = \( 2\cdot1\cdot1 \)
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 31737.2.a.a

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)

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

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 9408
\( \Gamma_0(N) \)-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])

\( L^{(3)}(E,1)/3! \) ≈ \( 4.21629322542 \)

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\((j)_{-}\)
\(3\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(71\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(149\) \(1\) \( I_{1} \) 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 71 149
Reduction type ss nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary nonsplit nonsplit
$\lambda$-invariant(s) 3,4 3 3 3 3 3 3 5 3 3 3 3 5 3 3 3 3
$\mu$-invariant(s) 0,0 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 31737a 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 3.1.42316.1 \(\Z/2\Z\) Not in database
6 6.0.18943221352624.1 \(\Z/2\Z \times \Z/2\Z\) 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.