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([0,0]),K([1,-1]),K([1,1]),K([-1,-2]),K([1,0])])
gp: E = ellinit([Polrev([0,0]),Polrev([1,-1]),Polrev([1,1]),Polrev([-1,-2]),Polrev([1,0])], K);
magma: E := EllipticCurve([K![0,0],K![1,-1],K![1,1],K![-1,-2],K![1,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-4a-5)\) | = | \((3,a+2)^{3}\cdot(5,a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 135 \) | = | \(3^{3}\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((5a+175)\) | = | \((3,a+2)^{5}\cdot(5,a)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -30375 \) | = | \(-3^{5}\cdot5^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{65536}{25} a - \frac{32768}{5} \) | ||
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 | $\left(-1 : 1 : 1\right)$ |
| Height | \(0.21375163945608973626657122498786902918\) |
| 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: | \( 0.21375163945608973626657122498786902918 \) | ||
| Period: | \( 26.857933775845776690992336928600160156 \) | ||
| Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.8154406392904912180579257449083019938 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((3,a+2)\) | \(3\) | \(1\) | \(IV\) | Additive | \(-1\) | \(3\) | \(5\) | \(0\) |
| \((5,a)\) | \(5\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
135.3-e
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.