Base field \(\Q(\sqrt{15}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 15 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-15, 0, 1]))
gp: K = nfinit(Polrev([-15, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-15, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([0,0]),K([-102668,-26498]),K([-17913888,-4625282])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,-1]),Polrev([0,0]),Polrev([-102668,-26498]),Polrev([-17913888,-4625282])], K);
magma: E := EllipticCurve([K![1,1],K![0,-1],K![0,0],K![-102668,-26498],K![-17913888,-4625282]]);
This is not a global minimal model: it is minimal at all primes except \((2,a+1)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((5a-15)\) | = | \((2,a+1)\cdot(3,a)\cdot(5,a)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 150 \) | = | \(2\cdot3\cdot5^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((15116544000)\) | = | \((2,a+1)^{22}\cdot(3,a)^{20}\cdot(5,a)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 228509902503936000000 \) | = | \(2^{22}\cdot3^{20}\cdot5^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
Minimal discriminant: | \((236196000)\) | = | \((2,a+1)^{10}\cdot(3,a)^{20}\cdot(5,a)^{6}\) |
Minimal discriminant norm: | \( 55788550416000000 \) | = | \(2^{10}\cdot3^{20}\cdot5^{6}\) |
j-invariant: | \( \frac{502270291349}{1889568} \) | ||
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 | $\left(-36 a - 128 : 37 a + 199 : 1\right)$ |
Height | \(0.34320248968635121134245445023144194223\) |
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: | $\left(-\frac{63}{2} a - 128 : \frac{319}{4} a + \frac{1201}{4} : 1\right)$ |
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: | \( 0.34320248968635121134245445023144194223 \) | ||
Period: | \( 0.79245572393389839095552774244137993143 \) | ||
Tamagawa product: | \( 160 \) = \(2\cdot( 2^{2} \cdot 5 )\cdot2^{2}\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.8089227668550712640329538210240554786 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a+1)\) | \(2\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
\((3,a)\) | \(3\) | \(20\) | \(I_{20}\) | Split multiplicative | \(-1\) | \(1\) | \(20\) | \(20\) |
\((5,a)\) | \(5\) | \(4\) | \(I_0^{*}\) | Additive | \(1\) | \(2\) | \(6\) | \(0\) |
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 |
\(5\) | 5B.4.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
150.1-c
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 450.b1 |
\(\Q\) | 1200.m1 |