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([1,1]),K([-1,0]),K([0,1]),K([-14281,-4515]),K([912811,288657])])
gp: E = ellinit([Polrev([1,1]),Polrev([-1,0]),Polrev([0,1]),Polrev([-14281,-4515]),Polrev([912811,288657])], K);
magma: E := EllipticCurve([K![1,1],K![-1,0],K![0,1],K![-14281,-4515],K![912811,288657]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((18,a+8)\) | = | \((2,a)\cdot(3,a+2)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 18 \) | = | \(2\cdot3^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((24448a-108544)\) | = | \((2,a)^{15}\cdot(3,a+2)^{11}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 5804752896 \) | = | \(2^{15}\cdot3^{11}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{14810400553709}{62208} a - \frac{11708649294791}{15552} \) | ||
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 | $\left(16 a + 51 : -48 a - 150 : 1\right)$ |
| Height | \(0.16714868475056400815434407510027208386\) |
| 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: | \( 0.16714868475056400815434407510027208386 \) | ||
| Period: | \( 4.7092788325795081011016180663285873990 \) | ||
| Tamagawa product: | \( 4 \) = \(1\cdot2^{2}\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 0.99567444428320510111002253842512457306 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(1\) | \(I_{15}\) | Non-split multiplicative | \(1\) | \(1\) | \(15\) | \(15\) |
| \((3,a+2)\) | \(3\) | \(4\) | \(I_{5}^{*}\) | Additive | \(-1\) | \(2\) | \(11\) | \(5\) |
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 |
| \(5\) | 5B.4.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5 and 15.
Its isogeny class
18.2-b
consists of curves linked by isogenies of
degrees dividing 15.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.