Base field \(\Q(\sqrt{97}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 24 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-24, -1, 1]))
gp: K = nfinit(Polrev([-24, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-24, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([1,1]),K([1,0]),K([-943641,173964]),K([398131134,-73395952])])
gp: E = ellinit([Polrev([1,0]),Polrev([1,1]),Polrev([1,0]),Polrev([-943641,173964]),Polrev([398131134,-73395952])], K);
magma: E := EllipticCurve([K![1,0],K![1,1],K![1,0],K![-943641,173964],K![398131134,-73395952]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((5a-27)\) | = | \((-7a-31)\cdot(-2a-9)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 6 \) | = | \(2\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((-377a+4497)\) | = | \((-7a-31)^{8}\cdot(-2a-9)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 15116544 \) | = | \(2^{8}\cdot3^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( -\frac{2521778623}{15116544} a + \frac{9408883739}{1889568} \) | ||
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: | \(0\) | |
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: | $\left(\frac{601}{4} a - \frac{3269}{4} : -\frac{601}{8} a + \frac{3265}{8} : 1\right)$ | $\left(-\frac{405}{4} a + 547 : \frac{405}{8} a - 274 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
|
|||
Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 17.393001304543770351325901290564998833 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 0.44149792936831376246003588675230703709 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-7a-31)\) | \(2\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
\((-2a-9)\) | \(3\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
6.3-a
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.