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([1,1]),K([0,0]),K([0,1]),K([9988013356524,1454820719431]),K([5884621003953910568,857134272549098597])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,0]),Polrev([0,1]),Polrev([9988013356524,1454820719431]),Polrev([5884621003953910568,857134272549098597])], K);
magma: E := EllipticCurve([K![1,1],K![0,0],K![0,1],K![9988013356524,1454820719431],K![5884621003953910568,857134272549098597]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-275a-1888)\) | = | \((-a+8)\cdot(-52a+409)\) |
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: | \((263a-2072)\) | = | \((-a+8)\cdot(-52a+409)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 13122 \) | = | \(2\cdot3^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1333879}{13122} a + \frac{368465}{486} \) | ||
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: | 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: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 16.229875353437826996502529645015272289 \) | ||
| Tamagawa product: | \( 8 \) = \(1\cdot2^{3}\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 8.8140458327149188481878645269569738091 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+8)\) | \(2\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((-52a+409)\) | \(3\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
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 6.2-a 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.