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([1,0]),K([0,0]),K([0,0]),K([-81,0]),K([6561,0])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([0,0]),Polrev([-81,0]),Polrev([6561,0])], K);
magma: E := EllipticCurve([K![1,0],K![0,0],K![0,0],K![-81,0],K![6561,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-20a+10)\) | = | \((-a+2)\cdot(2)\cdot(-a)\cdot(-a+1)\cdot(a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 2100 \) | = | \(3\cdot4\cdot5\cdot5\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((-18600435000)\) | = | \((-a+2)^{24}\cdot(2)^{3}\cdot(-a)^{4}\cdot(-a+1)^{4}\cdot(a+3)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 345976182189225000000 \) | = | \(3^{24}\cdot4^{3}\cdot5^{4}\cdot5^{4}\cdot7^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{58818484369}{18600435000} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ |
Mordell-Weil group
Rank: | \(1\) |
Generator | $\left(-54 a + 135 : 783 a - 2241 : 1\right)$ |
Height | \(0.72590477277517502910895194138309555762\) |
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: | $\left(36 : 207 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.72590477277517502910895194138309555762 \) | ||
Period: | \( 0.99044778118035162339693268445349679217 \) | ||
Tamagawa product: | \( 576 \) = \(( 2^{3} \cdot 3 )\cdot3\cdot2\cdot2\cdot2\) | ||
Torsion order: | \(6\) | ||
Leading coefficient: | \( 5.0205531170414172831215845522919499719 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a+2)\) | \(3\) | \(24\) | \(I_{24}\) | Split multiplicative | \(-1\) | \(1\) | \(24\) | \(24\) |
\((2)\) | \(4\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
\((-a)\) | \(5\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
\((-a+1)\) | \(5\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
\((a+3)\) | \(7\) | \(2\) | \(I_{2}\) | 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 |
\(3\) | 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
2100.1-d
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 |
---|---|
\(\Q\) | 210.d6 |
\(\Q\) | 4410.f6 |