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([0,1,0,0]),K([5,-2,-1,0]),K([-1,-5,1,1]),K([18,-97,6,15]),K([21,-165,12,26])])
gp: E = ellinit([Polrev([0,1,0,0]),Polrev([5,-2,-1,0]),Polrev([-1,-5,1,1]),Polrev([18,-97,6,15]),Polrev([21,-165,12,26])], K);
magma: E := EllipticCurve([K![0,1,0,0],K![5,-2,-1,0],K![-1,-5,1,1],K![18,-97,6,15],K![21,-165,12,26]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3+a^2-5a+1)\) | = | \((a^3+a^2-5a+1)\) |
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: | \((2a^3-a^2-9a+10)\) | = | \((a^3+a^2-5a+1)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -1024 \) | = | \(-2^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{13902918145}{1024} a^{3} + \frac{8433777799}{512} a^{2} - \frac{59975003189}{1024} a + \frac{6307656687}{1024} \) | ||
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 | $\left(a^{3} + a^{2} - 5 a - 1 : -3 a^{3} - 2 a^{2} + 18 a - 1 : 1\right)$ |
| Height | \(0.095999855190781272982368037317809530287\) |
| 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: | $\left(\frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{5}{4} a - \frac{5}{4} : -\frac{3}{4} a^{3} - \frac{3}{4} a^{2} + \frac{35}{8} a + \frac{3}{8} : 1\right)$ |
| 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: | \( 0.095999855190781272982368037317809530287 \) | ||
| Period: | \( 376.96093132674340912743245500978118826 \) | ||
| Tamagawa product: | \( 10 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.72709064112431 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3+a^2-5a+1)\) | \(2\) | \(10\) | \(I_{10}\) | Split multiplicative | \(-1\) | \(1\) | \(10\) | \(10\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
2.1-b
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.