Base field 6.6.1229312.1
Generator \(a\), with minimal polynomial \( x^{6} - 10 x^{4} + 24 x^{2} - 8 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-8, 0, 24, 0, -10, 0, 1]))
gp: K = nfinit(Polrev([-8, 0, 24, 0, -10, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 0, 24, 0, -10, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,1,1/2,0,0,0]),K([-2,2,2,-2,-1/4,1/4]),K([-2,-3,1/2,1/2,0,0]),K([-68,95,37,-46,-17/4,19/4]),K([254,-414,-243/2,199,49/4,-21])])
gp: E = ellinit([Polrev([-2,1,1/2,0,0,0]),Polrev([-2,2,2,-2,-1/4,1/4]),Polrev([-2,-3,1/2,1/2,0,0]),Polrev([-68,95,37,-46,-17/4,19/4]),Polrev([254,-414,-243/2,199,49/4,-21])], K);
magma: E := EllipticCurve([K![-2,1,1/2,0,0,0],K![-2,2,2,-2,-1/4,1/4],K![-2,-3,1/2,1/2,0,0],K![-68,95,37,-46,-17/4,19/4],K![254,-414,-243/2,199,49/4,-21]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/4a^5-2a^3+3a)\) | = | \((1/4a^5-2a^3+3a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 8 \) | = | \(8\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-1/4a^5+2a^3-3a)\) | = | \((1/4a^5-2a^3+3a)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 8 \) | = | \(8\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{753047}{8} a^{5} - 753047 a^{3} + \frac{2259141}{2} a - 574009 \) | ||
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: | \(\Z/3\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(\frac{1}{4} a^{4} + \frac{1}{2} a^{3} - 2 a^{2} - 2 a + 5 : -\frac{1}{4} a^{5} + \frac{3}{4} a^{4} + 2 a^{3} - \frac{13}{2} a^{2} - 2 a + 11 : 1\right)$ |
| 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: | \( 9672.8917555108039030919381529492494567 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 0.969355 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/4a^5-2a^3+3a)\) | \(8\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.1 |
| \(7\) | 7B.6.1[3] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 7 and 21.
Its isogeny class
8.1-c
consists of curves linked by isogenies of
degrees dividing 21.
Base change
This elliptic curve is not a \(\Q\)-curve. It is the base change of the following elliptic curve:
| Base field | Curve |
|---|---|
| \(\Q(\sqrt{2}) \) | 2.2.8.1-98.2-b2 |