Base field \(\Q(\sqrt{229}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 57 \); class number \(3\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-57, -1, 1]))
gp: K = nfinit(Polrev([-57, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-57, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([0,-1]),K([1,1]),K([-78,-26]),K([273,1])])
gp: E = ellinit([Polrev([0,1]),Polrev([0,-1]),Polrev([1,1]),Polrev([-78,-26]),Polrev([273,1])], K);
magma: E := EllipticCurve([K![0,1],K![0,-1],K![1,1],K![-78,-26],K![273,1]]);
This is not a global minimal model: it is minimal at all primes except \((3,a)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2)\) | = | \((2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 4 \) | = | \(4\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((584a-3612)\) | = | \((3,a)^{12}\cdot(2)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 8503056 \) | = | \(3^{12}\cdot4^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
Minimal discriminant: | \((4)\) | = | \((2)^{2}\) |
Minimal discriminant norm: | \( 16 \) | = | \(4^{2}\) |
j-invariant: | \( -\frac{39151}{4} a + \frac{326425}{4} \) | ||
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 | $\left(\frac{4}{9} a - \frac{11}{3} : \frac{62}{27} a - \frac{43}{9} : 1\right)$ |
Height | \(0.50385517527508409140994503808477490977\) |
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: | \( 0.50385517527508409140994503808477490977 \) | ||
Period: | \( 29.615381931712035886116083401693151882 \) | ||
Tamagawa product: | \( 2 \) = \(1\cdot2\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 3.9442579695526164542765343937068125060 \) | ||
Analytic order of Ш: | \( 1 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
Primes of good reduction for the curve but which divide the
discriminant of the model above (if any) are included.
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((3,a)\) | \(3\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
\((2)\) | \(4\) | \(2\) | \(I_{2}\) | 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 |
---|---|
\(2\) | 2Cn |
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 4.1-c 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.