Base field 5.5.195829.1
Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 7 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 7, 6, -5, -2, 1]))
gp: K = nfinit(Polrev([1, 7, 6, -5, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 6, -5, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-1,1,0,0]),K([2,1,-1,0,0]),K([-4,-9,4,3,-1]),K([27,32,-19,-10,4]),K([78,95,-60,-30,13])])
gp: E = ellinit([Polrev([-1,-1,1,0,0]),Polrev([2,1,-1,0,0]),Polrev([-4,-9,4,3,-1]),Polrev([27,32,-19,-10,4]),Polrev([78,95,-60,-30,13])], K);
magma: E := EllipticCurve([K![-1,-1,1,0,0],K![2,1,-1,0,0],K![-4,-9,4,3,-1],K![27,32,-19,-10,4],K![78,95,-60,-30,13]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^4-2a^3-5a^2+7a+5)\) | = | \((a^4-2a^3-5a^2+7a+5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 2 \) | = | \(2\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-3a^4+11a^3+a^2-20a-29)\) | = | \((a^4-2a^3-5a^2+7a+5)^{24}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -16777216 \) | = | \(-2^{24}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{140854061737}{16777216} a^{4} + \frac{521974831409}{16777216} a^{3} - \frac{86178293541}{8388608} a^{2} - \frac{34744410679}{1048576} a - \frac{76866520719}{16777216} \) | ||
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 | $\left(a^{3} - a^{2} - 3 a + 2 : -4 a^{4} + 7 a^{3} + 16 a^{2} - 22 a - 14 : 1\right)$ |
| Height | \(0.31108971388969395128795013368176123704\) |
| Torsion structure: | \(\Z/3\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(0 : 2 a^{4} - 5 a^{3} - 9 a^{2} + 16 a + 12 : 1\right)$ |
| 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: | \( 0.31108971388969395128795013368176123704 \) | ||
| Period: | \( 2223.4146055255491789990943609551893152 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 1.73670115 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-2a^3-5a^2+7a+5)\) | \(2\) | \(2\) | \(I_{24}\) | Non-split multiplicative | \(1\) | \(1\) | \(24\) | \(24\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
2.1-b
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.