Base field \(\Q(\sqrt{-7}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))
gp: K = nfinit(Polrev([2, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([1,0]),K([-3,0]),K([1,0])])
gp: E = ellinit([Polrev([1,0]),Polrev([1,0]),Polrev([1,0]),Polrev([-3,0]),Polrev([1,0])], K);
magma: E := EllipticCurve([K![1,0],K![1,0],K![1,0],K![-3,0],K![1,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((50)\) | = | \((a)\cdot(-a+1)\cdot(5)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 2500 \) | = | \(2\cdot2\cdot25^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-800)\) | = | \((a)^{5}\cdot(-a+1)^{5}\cdot(5)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 640000 \) | = | \(2^{5}\cdot2^{5}\cdot25^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{121945}{32} \) | ||
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 | $\left(\frac{1}{4} a + \frac{1}{4} : -\frac{3}{8} a + \frac{1}{8} : 1\right)$ |
| Height | \(0.73454693764387876804439717843432206178\) |
| Torsion structure: | \(\Z/5\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-1 : -2 : 1\right)$ |
| 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: | \( 0.73454693764387876804439717843432206178 \) | ||
| Period: | \( 4.2725002404569654263643326935944817542 \) | ||
| Tamagawa product: | \( 25 \) = \(5\cdot5\cdot1\) | ||
| Torsion order: | \(5\) | ||
| Leading coefficient: | \( 4.7447421903725299910957781957909473626 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
| \((-a+1)\) | \(2\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
| \((5)\) | \(25\) | \(1\) | \(II\) | Additive | \(1\) | \(2\) | \(2\) | \(0\) |
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.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5 and 15.
Its isogeny class
2500.2-b
consists of curves linked by isogenies of
degrees dividing 15.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 50.b3 |
| \(\Q\) | 2450.bd3 |