Base field 4.4.9792.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 7 x^{2} + 2 x + 7 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([7, 2, -7, -2, 1]))
gp: K = nfinit(Polrev([7, 2, -7, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, 2, -7, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0,0]),K([-4,-2,1,0]),K([0,0,0,0]),K([-7,8,5,-2]),K([16,-6,-11,1])])
gp: E = ellinit([Polrev([1,1,0,0]),Polrev([-4,-2,1,0]),Polrev([0,0,0,0]),Polrev([-7,8,5,-2]),Polrev([16,-6,-11,1])], K);
magma: E := EllipticCurve([K![1,1,0,0],K![-4,-2,1,0],K![0,0,0,0],K![-7,8,5,-2],K![16,-6,-11,1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-a-1)\) | = | \((a^3-3a^2-3a+4)\cdot(a^3-3a^2-2a+5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 36 \) | = | \(4\cdot9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((9a^3-27a^2-45a+72)\) | = | \((a^3-3a^2-3a+4)\cdot(a^3-3a^2-2a+5)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -236196 \) | = | \(-4\cdot9^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{7769042107}{27} a^{3} - \frac{4821837803}{18} a^{2} - \frac{124228493479}{54} a - \frac{101732420597}{54} \) | ||
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 | $\left(-\frac{2417}{1681} a^{3} + \frac{9978}{1681} a^{2} + \frac{7734}{1681} a - \frac{13976}{1681} : -\frac{23586}{68921} a^{3} - \frac{606739}{68921} a^{2} - \frac{151310}{68921} a + \frac{767595}{68921} : 1\right)$ |
| Height | \(1.7434957495724121427232140082824254313\) |
| 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: | $\left(a^{3} - 3 a^{2} - 3 a + 7 : 3 a^{3} - 11 a^{2} - 10 a + 18 : 1\right)$ |
| 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: | \( 1.7434957495724121427232140082824254313 \) | ||
| Period: | \( 226.51641703827709526328868087373898421 \) | ||
| Tamagawa product: | \( 5 \) = \(1\cdot5\) | ||
| Torsion order: | \(5\) | ||
| Leading coefficient: | \( 3.19282313038363 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-3a^2-3a+4)\) | \(4\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((a^3-3a^2-2a+5)\) | \(9\) | \(5\) | \(I_{5}\) | 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
36.1-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.