Base field 4.4.16448.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 6 x^{2} + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, -6, -2, 1]))
gp: K = nfinit(Polrev([2, 0, -6, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, -6, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([3,-3,-3,1]),K([2,-4,-3,1]),K([2,-3,-3,1]),K([-230,-441,-99,66]),K([1912,3510,750,-516])])
gp: E = ellinit([Polrev([3,-3,-3,1]),Polrev([2,-4,-3,1]),Polrev([2,-3,-3,1]),Polrev([-230,-441,-99,66]),Polrev([1912,3510,750,-516])], K);
magma: E := EllipticCurve([K![3,-3,-3,1],K![2,-4,-3,1],K![2,-3,-3,1],K![-230,-441,-99,66],K![1912,3510,750,-516]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-3a^2-3a+4)\) | = | \((a)\cdot(-2a^3+5a^2+9a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 10 \) | = | \(2\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((6a^3-18a^2-18a+4)\) | = | \((a)^{5}\cdot(-2a^3+5a^2+9a-3)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -20000 \) | = | \(-2^{5}\cdot5^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{100210679}{2500} a^{3} - \frac{90943874}{625} a^{2} - \frac{18207153}{625} a + \frac{79650407}{1250} \) | ||
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 | $\left(2 a^{3} - 5 a^{2} - 9 a + 4 : -a^{3} + 14 a^{2} - 21 a - 62 : 1\right)$ |
| Height | \(0.0055026159067906362676745996560631418067\) |
| 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: | \( 0.0055026159067906362676745996560631418067 \) | ||
| Period: | \( 2194.3348998712322602743735051083092072 \) | ||
| Tamagawa product: | \( 10 \) = \(5\cdot2\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.76595869494288 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
| \((-2a^3+5a^2+9a-3)\) | \(5\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
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 10.1-a 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.