Base field \(\Q(\sqrt{17}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 4 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-4, -1, 1]))
gp: K = nfinit(Polrev([-4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([-1,-1]),K([1,0]),K([-15,7]),K([27,-11])])
gp: E = ellinit([Polrev([1,0]),Polrev([-1,-1]),Polrev([1,0]),Polrev([-15,7]),Polrev([27,-11])], K);
magma: E := EllipticCurve([K![1,0],K![-1,-1],K![1,0],K![-15,7],K![27,-11]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-20a-22)\) | = | \((-a+2)\cdot(-a-1)\cdot(-2a+3)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 676 \) | = | \(2\cdot2\cdot13^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((28a-848)\) | = | \((-a+2)^{10}\cdot(-a-1)^{2}\cdot(-2a+3)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 692224 \) | = | \(2^{10}\cdot2^{2}\cdot13^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{7851401}{1024} a - \frac{20220829}{1024} \) | ||
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 | $\left(3 : -a - 2 : 1\right)$ |
| Height | \(0.015453254529392409587955364724252069252\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.015453254529392409587955364724252069252 \) | ||
| Period: | \( 21.380664755058126118860172414124236729 \) | ||
| Tamagawa product: | \( 20 \) = \(( 2 \cdot 5 )\cdot2\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.2053591100327636568304092929083371963 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+2)\) | \(2\) | \(10\) | \(I_{10}\) | Split multiplicative | \(-1\) | \(1\) | \(10\) | \(10\) |
| \((-a-1)\) | \(2\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((-2a+3)\) | \(13\) | \(1\) | \(II\) | Additive | \(1\) | \(2\) | \(2\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 676.4-h consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.