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([1,1]),K([1,-1]),K([0,1]),K([-4082,-2269]),K([19299,10782])])
gp: E = ellinit([Polrev([1,1]),Polrev([1,-1]),Polrev([0,1]),Polrev([-4082,-2269]),Polrev([19299,10782])], K);
magma: E := EllipticCurve([K![1,1],K![1,-1],K![0,1],K![-4082,-2269],K![19299,10782]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-10a+20)\) | = | \((-a+2)\cdot(2)\cdot(-a)\cdot(-a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 300 \) | = | \(3\cdot4\cdot5\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((5859375000)\) | = | \((-a+2)^{2}\cdot(2)^{3}\cdot(-a)^{12}\cdot(-a+1)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 34332275390625000000 \) | = | \(3^{2}\cdot4^{3}\cdot5^{12}\cdot5^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{10316097499609}{5859375000} \) | ||
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 | $\left(-53 a + 143 : -894 a + 2186 : 1\right)$ |
| Height | \(0.18070390582850248052720280734757793316\) |
| Torsion structure: | \(\Z/6\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-3 a - 7 : 106 a + 186 : 1\right)$ |
| 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: | \( 0.18070390582850248052720280734757793316 \) | ||
| Period: | \( 1.3418722835546968520385922310871675988 \) | ||
| Tamagawa product: | \( 864 \) = \(2\cdot3\cdot( 2^{2} \cdot 3 )\cdot( 2^{2} \cdot 3 )\) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 2.5398631222515453047639516748972027028 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+2)\) | \(3\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((2)\) | \(4\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
| \((-a)\) | \(5\) | \(12\) | \(I_{12}\) | Split multiplicative | \(-1\) | \(1\) | \(12\) | \(12\) |
| \((-a+1)\) | \(5\) | \(12\) | \(I_{12}\) | Split multiplicative | \(-1\) | \(1\) | \(12\) | \(12\) |
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.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
300.1-j
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 90.c2 |
| \(\Q\) | 1470.d2 |