Base field \(\Q(\sqrt{10}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 10 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-10, 0, 1]))
gp: K = nfinit(Polrev([-10, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([-1,-1]),K([0,1]),K([-155,-47]),K([-900,-280])])
gp: E = ellinit([Polrev([0,1]),Polrev([-1,-1]),Polrev([0,1]),Polrev([-155,-47]),Polrev([-900,-280])], K);
magma: E := EllipticCurve([K![0,1],K![-1,-1],K![0,1],K![-155,-47],K![-900,-280]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-6a)\) | = | \((2,a)^{3}\cdot(3,a+1)\cdot(3,a+2)\cdot(5,a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 360 \) | = | \(2^{3}\cdot3\cdot3\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((100800a+1778400)\) | = | \((2,a)^{10}\cdot(3,a+1)^{2}\cdot(3,a+2)^{12}\cdot(5,a)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 3061100160000 \) | = | \(2^{10}\cdot3^{2}\cdot3^{12}\cdot5^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{74541355592}{13286025} a + \frac{260191879754}{13286025} \) | ||
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 | $\left(-2 a - 4 : 3 a + 10 : 1\right)$ | |
| Height | \(1.9153761138270216799532869080342915066\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(-a - \frac{15}{2} : \frac{13}{4} a + 5 : 1\right)$ | $\left(-a - 5 : 2 a + 5 : 1\right)$ |
| 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: | \( 1.9153761138270216799532869080342915066 \) | ||
| Period: | \( 3.4156527182109690912663256523742937128 \) | ||
| Tamagawa product: | \( 32 \) = \(2\cdot2\cdot2\cdot2^{2}\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 4.1376882947345481013840900671470665245 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(2\) | \(III^{*}\) | Additive | \(-1\) | \(3\) | \(10\) | \(0\) |
| \((3,a+1)\) | \(3\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((3,a+2)\) | \(3\) | \(2\) | \(I_{12}\) | Non-split multiplicative | \(1\) | \(1\) | \(12\) | \(12\) |
| \((5,a)\) | \(5\) | \(4\) | \(I_{4}\) | 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 and 4.
Its isogeny class
360.1-d
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.