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([1,1]),K([0,-1]),K([1,0]),K([-578807959,-130820947]),K([8001269927498,1808429993272])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,-1]),Polrev([1,0]),Polrev([-578807959,-130820947]),Polrev([8001269927498,1808429993272])], K);
magma: E := EllipticCurve([K![1,1],K![0,-1],K![1,0],K![-578807959,-130820947],K![8001269927498,1808429993272]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a-6)\) | = | \((-7a+38)\cdot(-2a-9)\) |
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: | \((48877a-265224)\) | = | \((-7a+38)^{8}\cdot(-2a-9)^{11}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 45349632 \) | = | \(2^{8}\cdot3^{11}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{5834396582701}{45349632} a - \frac{25805666462171}{45349632} \) | ||
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 | $\left(2237 a + 9895 : -42446 a - 187804 : 1\right)$ |
| Height | \(0.011649964372140303261926204341734899154\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.011649964372140303261926204341734899154 \) | ||
| Period: | \( 12.706965565921545003013270154417378919 \) | ||
| Tamagawa product: | \( 22 \) = \(2\cdot11\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 0.66135289598110811755294180575376231710 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-7a+38)\) | \(2\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
| \((-2a-9)\) | \(3\) | \(11\) | \(I_{11}\) | Split multiplicative | \(-1\) | \(1\) | \(11\) | \(11\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 6.1-b consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.