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([0,0]),K([-266,-68]),K([-1784,-506])])
gp: E = ellinit([Polrev([0,0]),Polrev([-1,-1]),Polrev([0,0]),Polrev([-266,-68]),Polrev([-1784,-506])], K);
magma: E := EllipticCurve([K![0,0],K![-1,-1],K![0,0],K![-266,-68],K![-1784,-506]]);
This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-3a+15)\) | = | \((3,a+1)^{2}\cdot(3,a+2)\cdot(5,a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 135 \) | = | \(3^{2}\cdot3\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((696000a-1032000)\) | = | \((2,a)^{12}\cdot(3,a+1)^{9}\cdot(3,a+2)\cdot(5,a)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 3779136000000 \) | = | \(2^{12}\cdot3^{9}\cdot3\cdot5^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| Minimal discriminant: | \((10875a-16125)\) | = | \((3,a+1)^{9}\cdot(3,a+2)\cdot(5,a)^{6}\) |
| Minimal discriminant norm: | \( 922640625 \) | = | \(3^{9}\cdot3\cdot5^{6}\) |
| j-invariant: | \( -\frac{66006784}{375} a + \frac{226222784}{375} \) | ||
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 | $\left(-\frac{40}{9} a + \frac{73}{18} : -\frac{1325}{108} a + \frac{1925}{54} : 1\right)$ |
| Height | \(2.4917886595952619109030459294972334088\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(5 a + 11 : 0 : 1\right)$ |
| 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: | \( 2.4917886595952619109030459294972334088 \) | ||
| Period: | \( 3.6098533029722432540925705659458719723 \) | ||
| Tamagawa product: | \( 4 \) = \(1\cdot2\cdot1\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.8444660747057187919392215700267920443 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
Primes of good reduction for the curve but which divide the
discriminant of the model above (if any) are included.
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((3,a+1)\) | \(3\) | \(2\) | \(III^{*}\) | Additive | \(1\) | \(2\) | \(9\) | \(0\) |
| \((3,a+2)\) | \(3\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((5,a)\) | \(5\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
135.2-g
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.