Base field 4.4.17609.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + 10 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 10, -7, -1, 1]))
gp: K = nfinit(Polrev([-1, 10, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 10, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-5,1,1]),K([2,-6,0,1]),K([-1,-5,1,1]),K([-922,8193,-2285,-1895]),K([-40395,385907,-108595,-89471])])
gp: E = ellinit([Polrev([-1,-5,1,1]),Polrev([2,-6,0,1]),Polrev([-1,-5,1,1]),Polrev([-922,8193,-2285,-1895]),Polrev([-40395,385907,-108595,-89471])], K);
magma: E := EllipticCurve([K![-1,-5,1,1],K![2,-6,0,1],K![-1,-5,1,1],K![-922,8193,-2285,-1895],K![-40395,385907,-108595,-89471]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-7a+6)\) | = | \((a^3+a^2-5a+1)^{2}\cdot(-a^3+6a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 28 \) | = | \(2^{2}\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-25a^3-18a^2+140a+5)\) | = | \((a^3+a^2-5a+1)^{4}\cdot(-a^3+6a-2)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -1882384 \) | = | \(-2^{4}\cdot7^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{58991001474401370845134}{117649} a^{3} - \frac{28633695056137501631935}{117649} a^{2} + \frac{370404780368043596995741}{117649} a - \frac{39714126185228454891650}{117649} \) | ||
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: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(18 a^{3} + \frac{87}{4} a^{2} - \frac{311}{4} a + \frac{17}{2} : -\frac{67}{2} a^{3} - \frac{325}{8} a^{2} + \frac{579}{4} a - \frac{113}{8} : 1\right)$ |
| 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: | \( 7.1189385516217228590402904215090202357 \) | ||
| Tamagawa product: | \( 6 \) = \(1\cdot( 2 \cdot 3 )\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 0.724238596975327 \) | ||
| Analytic order of Ш: | \( 9 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3+a^2-5a+1)\) | \(2\) | \(1\) | \(IV\) | Additive | \(-1\) | \(2\) | \(4\) | \(0\) |
| \((-a^3+6a-2)\) | \(7\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
28.1-b
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.