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([0,0]),K([0,1]),K([0,-3]),K([9,-3])])
gp: E = ellinit([Polrev([0,1]),Polrev([0,0]),Polrev([0,1]),Polrev([0,-3]),Polrev([9,-3])], K);
magma: E := EllipticCurve([K![0,1],K![0,0],K![0,1],K![0,-3],K![9,-3]]);
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: | \((1)\) | = | \((1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 1 \) | = | 1 |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((146a-903)\) | = | \((3,a)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 531441 \) | = | \(3^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((1)\) | = | \((1)\) |
Minimal discriminant norm: | \( 1 \) | = | \( 1 \) |
j-invariant: | \( 888615 a + 6445522 \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ |
Mordell-Weil group
Rank: | \(0\) |
Torsion structure: | \(\Z/3\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(0 : -a + 3 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 38.582162938340324629972556315173651306 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(3\) | ||
Leading coefficient: | \( 2.5495810915444287630412314516586064507 \) | ||
Analytic order of Ш: | \( 9 \) (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 | \(1\) | \(0\) | \(0\) | \(0\) |
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 |
\(3\) | 3B.1.1 |
\(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5 and 15.
Its isogeny class
1.1-a
consists of curves linked by isogenies of
degrees dividing 15.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.