Base field 6.6.1279733.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 6 x^{4} + 10 x^{3} + 10 x^{2} - 11 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -11, 10, 10, -6, -2, 1]))
gp: K = nfinit(Polrev([-1, -11, 10, 10, -6, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -11, 10, 10, -6, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1,-3,-1,1,0]),K([2,-4,-6,4,2,-1]),K([0,-1,2,-3,-1,1]),K([-23,31,-2,-28,1,4]),K([31,-46,1,62,3,-12])])
gp: E = ellinit([Polrev([0,1,-3,-1,1,0]),Polrev([2,-4,-6,4,2,-1]),Polrev([0,-1,2,-3,-1,1]),Polrev([-23,31,-2,-28,1,4]),Polrev([31,-46,1,62,3,-12])], K);
magma: E := EllipticCurve([K![0,1,-3,-1,1,0],K![2,-4,-6,4,2,-1],K![0,-1,2,-3,-1,1],K![-23,31,-2,-28,1,4],K![31,-46,1,62,3,-12]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^5+a^4+5a^3-3a^2-5a+1)\) | = | \((a^4-2a^3-3a^2+6a-1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 49 \) | = | \(7^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2a^5+a^4-11a^3-5a^2+10a-9)\) | = | \((a^4-2a^3-3a^2+6a-1)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -823543 \) | = | \(-7^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{501548626}{7} a^{5} + \frac{166452530}{7} a^{4} - \frac{2621036258}{7} a^{3} - \frac{1096676327}{7} a^{2} + \frac{2456909923}{7} a + \frac{213445339}{7} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(2 a^{5} - 6 a^{4} - 16 a^{3} + 24 a^{2} + 45 a + 6 : 49 a^{5} - 38 a^{4} - 332 a^{3} + 106 a^{2} + 569 a + 48 : 1\right)$ |
| Height | \(0.15501657803529268392899076899770168364\) |
| Torsion structure: | \(\Z/7\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-a^{4} + a^{3} + 3 a^{2} - a + 2 : -2 a^{3} + a^{2} + 7 a : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.15501657803529268392899076899770168364 \) | ||
| Period: | \( 111773.58238196119533762142392370875699 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(7\) | ||
| Leading coefficient: | \( 3.75097 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-2a^3-3a^2+6a-1)\) | \(7\) | \(2\) | \(I_{1}^{*}\) | Additive | \(-1\) | \(2\) | \(7\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(7\) | 7B.1.1[3] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
49.1-a
consists of curves linked by isogenies of
degree 7.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.