Base field \(\Q(\sqrt{217}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 54 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-54, -1, 1]))
gp: K = nfinit(Polrev([-54, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-54, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-36,0]),K([-70,0])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,0]),Polrev([-36,0]),Polrev([-70,0])], K);
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![-36,0],K![-70,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-996a+7834)\) | = | \((-a+8)\cdot(-a-7)\cdot(-498a+3917)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 28 \) | = | \(2\cdot2\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((941192)\) | = | \((-a+8)^{3}\cdot(-a-7)^{3}\cdot(-498a+3917)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 885842380864 \) | = | \(2^{3}\cdot2^{3}\cdot7^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( \frac{4956477625}{941192} \) | ||
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 | $\left(\frac{2077}{144} : -\frac{5831}{864} a - \frac{7495}{1728} : 1\right)$ |
Height | \(4.9313153302169379460513800922809866265\) |
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: | $\left(-4 : -2 : 1\right)$ |
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: | \( 4.9313153302169379460513800922809866265 \) | ||
Period: | \( 3.9257159468709430520307637303495930307 \) | ||
Tamagawa product: | \( 12 \) = \(1\cdot1\cdot( 2^{2} \cdot 3 )\) | ||
Torsion order: | \(6\) | ||
Leading coefficient: | \( 0.87611379834049444913165700661958896547 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a+8)\) | \(2\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
\((-a-7)\) | \(2\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
\((-498a+3917)\) | \(7\) | \(12\) | \(I_{12}\) | 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 |
\(3\) | 3Cs.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
28.1-h
consists of curves linked by isogenies of
degrees dividing 18.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 14.a3 |
\(\Q\) | 94178.b3 |