Base field 4.4.4913.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 6 x^{2} + x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 1, -6, -1, 1]))
gp: K = nfinit(Polrev([1, 1, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0,0]),K([-1,0,0,0]),K([1,0,0,0]),K([-1,0,0,0]),K([0,0,0,0])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([-1,0,0,0]),Polrev([1,0,0,0]),Polrev([-1,0,0,0]),Polrev([0,0,0,0])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![-1,0,0,0],K![1,0,0,0],K![-1,0,0,0],K![0,0,0,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/2a^3-2a-3/2)\) | = | \((1/2a^3-2a-3/2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 17 \) | = | \(17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((17)\) | = | \((1/2a^3-2a-3/2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 83521 \) | = | \(17^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{35937}{17} \) | ||
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 | $\left(a^{3} - a^{2} - 5 a + 2 : -2 a^{3} + 2 a^{2} + 10 a - 5 : 1\right)$ | |
| Height | \(0.75881404443258304020801340081793683401\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/8\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(\frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{5}{4} a : -\frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{5}{8} a - \frac{1}{2} : 1\right)$ | $\left(\frac{3}{2} a^{3} - 2 a^{2} - 8 a + \frac{9}{2} : 3 a^{3} - 4 a^{2} - 17 a + 8 : 1\right)$ |
| 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: | \( 0.75881404443258304020801340081793683401 \) | ||
| Period: | \( 1466.5294063350350194979993636135137464 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(16\) | ||
| Leading coefficient: | \( 0.992276649128961 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/2a^3-2a-3/2)\) | \(17\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
17.1-a
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 4 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 17.a4 |
| \(\Q\) | 289.a4 |
| \(\Q(\sqrt{17}) \) | 2.2.17.1-17.1-a4 |
| \(\Q(\sqrt{17}) \) | 2.2.17.1-289.1-a4 |