Properties

Label 37.a1
Conductor \(37\)
Discriminant \(37\)
j-invariant \( \frac{110592}{37} \)
CM no
Rank \(1\)
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 1, -1, 0]); // or
magma: E := EllipticCurve("37a1");
sage: E = EllipticCurve([0, 0, 1, -1, 0]) # or
sage: E = EllipticCurve("37a1")
gp: E = ellinit([0, 0, 1, -1, 0]) \\ or
gp: E = ellinit("37a1")

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

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(0, 0\right) \)
\(\hat{h}(P)\) ≈  0.05111140824

Integral points

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

\( \left(-1, 0\right) \), \( \left(0, 0\right) \), \( \left(1, 0\right) \), \( \left(2, 2\right) \), \( \left(6, 14\right) \)

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
\( N \)  =  \( 37 \)  =  \(37\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
\(\Delta\)  =  \(37 \)  =  \(37 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
\(j \)  =  \( \frac{110592}{37} \)  =  \(2^{12} \cdot 3^{3} \cdot 37^{-1}\)
\( \text{End} (E) \)  =  \(\Z\)   (no Complex Multiplication)
\( \text{ST} (E) \)  =  $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
\( r \)  =  \(1\)
magma: Regulator(E);
sage: E.regulator()
\( \text{Reg} \)  ≈  \(0.05111140824\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
\( \Omega \)  ≈  \(5.98691729246\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
\( \prod_p c_p \)  =  \( 1 \)  = \( 1 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
\( \#E_{\text{tor}} \)  = \(1\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Ш\(_{\text{an}} \)  =   \(1\) (exact)

Modular invariants

Modular form 37.2.1.a

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

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

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
2 : curve is \( \Gamma_0(N) \)-optimal

Special L-value attached to the curve

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])

\( L'(E,1) \) ≈ \( 0.305999773834 \)

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\((j)_{-}\)
\(37\) \(1\) \( I_{1} \) 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

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

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

Isogenies

This curve has no rational isogenies. Its isogeny class 37.a consists of this curve only.

This is the elliptic curve of smallest conductor with rank 1.