Properties

Label 37.b3
Conductor 37
Discriminant 37
j-invariant \( \frac{4096000}{37} \)
CM no
Rank 0
Torsion Structure \(\Z/{3}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 1, -3, 1]); // or
magma: E := EllipticCurve("37b3");
sage: E = EllipticCurve([0, 1, 1, -3, 1]) # or
sage: E = EllipticCurve("37b3")
gp: E = ellinit([0, 1, 1, -3, 1]) \\ or
gp: E = ellinit("37b3")

\( y^2 + y = x^{3} + x^{2} - 3 x + 1 \)

Mordell-Weil group structure

\(\Z/{3}\Z\)

Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

\( \left(1, 0\right) \)

Integral points

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

\( \left(1, 0\right) \)

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 37 \)  =  \(37\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(37 \)  =  \(37 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{4096000}{37} \)  =  \(2^{15} \cdot 5^{3} \cdot 37^{-1}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
Rank: \(0\)
magma: Regulator(E);
sage: E.regulator()
Regulator: \(1\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(6.53112955743\)
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: \( 1 \)  = \( 1 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
Torsion order: \(3\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 37.2.a.b

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)

\( q + q^{3} - 2q^{4} - q^{7} - 2q^{9} + 3q^{11} - 2q^{12} - 4q^{13} + 4q^{16} + 6q^{17} + 2q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 6
\( \Gamma_0(N) \)-optimal: no
Manin constant: 3

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])

\( L(E,1) \) ≈ \( 0.725681061936 \)

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\((j)_{-}\)
\(37\) \(1\) \( I_{1} \) 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 \( p \) except those listed.

prime Image of Galois representation
\(3\) B.1.1

$p$-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 37
Reduction type ss ordinary split
$\lambda$-invariant(s) 0,5 0 1
$\mu$-invariant(s) 0,0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 37.b consists of 3 curves linked by isogenies of degrees dividing 9.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.3.148.1 \(\Z/6\Z\) Not in database
3.3.1369.1 \(\Z/9\Z\) 3.3.1369.1-37.1-a3
6 6.0.36963.1 \(\Z/9\Z\) Not in database
6.0.50602347.2 \(\Z/3\Z \times \Z/3\Z\) Not in database
6.6.810448.1 \(\Z/2\Z \times \Z/6\Z\) 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.

Additional information

The twist $E^{(-139)}$ of this curve $E$ by $\Q(\sqrt{-139})$ was the earliest elliptic curve proved to have analytic rank $3$, and thus the one first used to give an effective lower bound for the Gauss class number problem. See Proposition 7.4 and Theorem 8.1 in Gross and Zagier's paper in Invent. Math. 85 (1986). [The modularity theorem was not yet proved in general, but was known for $E$ by explicit calculation of the modular curve [$X_0(37)$]; the vanishing of the Heegner trace for discriminant $-139$ was proved by Zagier (Notices of the AMS 31 (1984), 739-743).] Soon after Oesterlé obtained several improvements on the constant factor in the bound, one of which was to replace the curve $E^{(-139)}$ of conductor $139^2 37 = 714877$ by the [minimal rank-$3$ curve] of conductor $5077$, whose modularity had recently been proved by Mestre, Oesterlé, and Serre. See pages 34-36 of [Goldfeld's exposition] in the 1985 Bull. AMS.