Base field \(\Q(\sqrt{21}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 5 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-5, -1, 1]))
gp: K = nfinit(Polrev([-5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,1]),K([0,1]),K([-1086,-512]),K([-21445,-11511])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,1]),Polrev([0,1]),Polrev([-1086,-512]),Polrev([-21445,-11511])], K);
magma: E := EllipticCurve([K![0,1],K![1,1],K![0,1],K![-1086,-512],K![-21445,-11511]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-10a+5)\) | = | \((-a+2)\cdot(-a)\cdot(-a+1)\cdot(a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 525 \) | = | \(3\cdot5\cdot5\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-205800a+900375)\) | = | \((-a+2)^{2}\cdot(-a)^{6}\cdot(-a+1)^{2}\cdot(a+3)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 413609765625 \) | = | \(3^{2}\cdot5^{6}\cdot5^{2}\cdot7^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{30507583626037513}{5359375} a + \frac{163944959727865366}{16078125} \) | ||
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 | $\left(-\frac{1498}{75} a + \frac{621}{25} : -\frac{116294}{1125} a + \frac{371539}{1125} : 1\right)$ | |
| Height | \(2.8845073792940830868045652058038541914\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(-10 a - 3 : 6 a + 25 : 1\right)$ | $\left(-\frac{21}{4} a - \frac{65}{4} : \frac{41}{4} a + \frac{105}{8} : 1\right)$ |
| 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: | \( 2.8845073792940830868045652058038541914 \) | ||
| Period: | \( 0.89498572943351444353142852255875947444 \) | ||
| Tamagawa product: | \( 48 \) = \(2\cdot2\cdot2\cdot( 2 \cdot 3 )\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 3.3800985900861339816524124033950246726 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+2)\) | \(3\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((-a)\) | \(5\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
| \((-a+1)\) | \(5\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((a+3)\) | \(7\) | \(6\) | \(I_{6}\) | 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 |
|---|---|
| \(2\) | 2Cs |
| \(3\) | 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
525.1-d
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.