Base field \(\Q(\sqrt{205}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 51 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-51, -1, 1]))
gp: K = nfinit(Polrev([-51, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-51, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([0,0]),K([0,0]),K([-1855392,-278636]),K([-1438058889,-215960088])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,0]),Polrev([0,0]),Polrev([-1855392,-278636]),Polrev([-1438058889,-215960088])], K);
magma: E := EllipticCurve([K![1,1],K![0,0],K![0,0],K![-1855392,-278636],K![-1438058889,-215960088]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((3a+21)\) | = | \((3,a)\cdot(3,a+2)\cdot(a+7)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 45 \) | = | \(3\cdot3\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((405)\) | = | \((3,a)^{4}\cdot(3,a+2)^{4}\cdot(a+7)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 164025 \) | = | \(3^{4}\cdot3^{4}\cdot5^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1114544804970241}{405} \) | ||
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 | $\left(-\frac{730}{9} a - \frac{4844}{9} : \frac{9472}{27} a + \frac{62963}{27} : 1\right)$ |
| Height | \(2.7060659344010859212576927062052306521\) |
| 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: | $\left(-\frac{323}{4} a - 541 : \frac{1405}{4} a + \frac{18637}{8} : 1\right)$ |
| 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: | \( 2.7060659344010859212576927062052306521 \) | ||
| Period: | \( 0.49042222054783514374841392352878611407 \) | ||
| Tamagawa product: | \( 32 \) = \(2^{2}\cdot2^{2}\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 5.9321422549211210555229969470370158599 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((3,a)\) | \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((3,a+2)\) | \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((a+7)\) | \(5\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4, 8 and 16.
Its isogeny class
45.1-b
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 75.b1 |
| \(\Q\) | 25215.f1 |