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([-1,0,1,0,0]),K([0,-4,-1,1,0]),K([-3,-9,3,3,-1]),K([-475,-557,376,178,-81]),K([-7409,-8623,5870,2756,-1268])])
gp: E = ellinit([Polrev([-1,0,1,0,0]),Polrev([0,-4,-1,1,0]),Polrev([-3,-9,3,3,-1]),Polrev([-475,-557,376,178,-81]),Polrev([-7409,-8623,5870,2756,-1268])], K);
magma: E := EllipticCurve([K![-1,0,1,0,0],K![0,-4,-1,1,0],K![-3,-9,3,3,-1],K![-475,-557,376,178,-81],K![-7409,-8623,5870,2756,-1268]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^4-2a^3-4a^2+6a+2)\) | = | \((a^4-2a^3-4a^2+6a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 11 \) | = | \(11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((6a^4-16a^3-26a^2+58a+39)\) | = | \((a^4-2a^3-4a^2+6a+2)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 161051 \) | = | \(11^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{106476089}{161051} a^{4} - \frac{198113042}{161051} a^{3} - \frac{361568756}{161051} a^{2} + \frac{44783789}{14641} a + \frac{9814598}{161051} \) | ||
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: | \(0\) |
| Torsion structure: | \(\Z/5\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(21 a^{4} - 46 a^{3} - 97 a^{2} + 144 a + 123 : 213 a^{4} - 462 a^{3} - 985 a^{2} + 1445 a + 1240 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 1769.4582863444200590879147921036046284 \) | ||
| Tamagawa product: | \( 5 \) | ||
| Torsion order: | \(5\) | ||
| Leading coefficient: | \( 0.799708701 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-2a^3-4a^2+6a+2)\) | \(11\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(5\) | 5B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
11.1-d
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.