This elliptic curve has everywhere good reduction and a rational point of order $37$.
Base field 6.6.300125.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 7, 2, -7, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 7*x^4 + 2*x^3 + 7*x^2 - 2*x - 1)
gp (2.8): K = nfinit(a^6 - a^5 - 7*a^4 + 2*a^3 + 7*a^2 - 2*a - 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([-2*a^5 + a^4 + 14*a^3 + 4*a^2 - 9*a - 2, -3*a^5 + 2*a^4 + 20*a^3 + 3*a^2 - 12*a, -5*a^5 + a^4 + 37*a^3 + 18*a^2 - 26*a - 7, 7*a^5 - a^4 - 51*a^3 - 28*a^2 + 27*a + 11, -13*a^5 + 4*a^4 + 92*a^3 + 40*a^2 - 56*a - 16]),K);
sage: E = EllipticCurve(K, [-2*a^5 + a^4 + 14*a^3 + 4*a^2 - 9*a - 2, -3*a^5 + 2*a^4 + 20*a^3 + 3*a^2 - 12*a, -5*a^5 + a^4 + 37*a^3 + 18*a^2 - 26*a - 7, 7*a^5 - a^4 - 51*a^3 - 28*a^2 + 27*a + 11, -13*a^5 + 4*a^4 + 92*a^3 + 40*a^2 - 56*a - 16])
gp (2.8): E = ellinit([-2*a^5 + a^4 + 14*a^3 + 4*a^2 - 9*a - 2, -3*a^5 + 2*a^4 + 20*a^3 + 3*a^2 - 12*a, -5*a^5 + a^4 + 37*a^3 + 18*a^2 - 26*a - 7, 7*a^5 - a^4 - 51*a^3 - 28*a^2 + 27*a + 11, -13*a^5 + 4*a^4 + 92*a^3 + 40*a^2 - 56*a - 16],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((1)\) | = | \((1)\) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 1 \) | = | 1 |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((1)\) | = | \((1)\) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 1 \) | = | 1 |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -9317 \) | ||
| magma: jInvariant(E);
sage: E.j_invariant()
gp (2.8): E.j
| ||||
| \( \text{End} (E) \) | = | \(\Z\) | (no Complex Multiplication ) | |
| magma: HasComplexMultiplication(E);
sage: E.has_cm(), E.cm_discriminant()
| ||||
| \( \text{ST} (E) \) | = | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
Rank not available.magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()
Regulator: not available
magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())
Torsion subgroup
| Structure: | \(\Z/37\Z\) |
|---|---|
| magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[2]
| |
| magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp (2.8): elltors(E)[1]
| |
| Generator: | $\left(0 : 6 a^{5} - a^{4} - 44 a^{3} - 23 a^{2} + 28 a + 9 : 1\right)$ |
| magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[3]
| |
Local data at primes of bad reduction
magma: LocalInformation(E);
sage: E.local_data()
No primes of bad reduction.
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(37\) | 37B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
37.
Its isogeny class
1.1-a
consists of curves linked by isogenies of
degree 37.
Base change
This curve is not the base-change of an elliptic curve defined over \(\Q\). It is a \(\Q\)-curve.Additional information
This elliptic curve has everywhere good reduction and a rational point of order $37$.