Base field 3.1.23.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, -1, 1]))
gp: K = nfinit(Polrev([1, 0, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0,0]),K([1,-1,0]),K([0,1,0]),K([-2357,3086,-1768]),K([-69423,92024,-52408])])
gp: E = ellinit([Polrev([0,0,0]),Polrev([1,-1,0]),Polrev([0,1,0]),Polrev([-2357,3086,-1768]),Polrev([-69423,92024,-52408])], K);
magma: E := EllipticCurve([K![0,0,0],K![1,-1,0],K![0,1,0],K![-2357,3086,-1768],K![-69423,92024,-52408]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-4a^2+7a+1)\) | = | \((2a^2-a)\cdot(3a^2-a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 259 \) | = | \(7\cdot37\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((64845a^2+287595a-308701)\) | = | \((2a^2-a)^{3}\cdot(3a^2-a+1)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -44576876749711411 \) | = | \(-7^{3}\cdot37^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{578622056111156353812664320}{44576876749711411} a^{2} + \frac{1015410575032007747136962560}{44576876749711411} a - \frac{766510266683364441673433088}{44576876749711411} \) | ||
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: | $\left(40 a^{2} - 68 a + 56 : 357 a^{2} - 631 a + 471 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 0.41554839776130810614684984214935377991 \) | ||
Tamagawa product: | \( 27 \) = \(3\cdot3^{2}\) | ||
Torsion order: | \(3\) | ||
Leading coefficient: | \( 0.51988698402933238686103269625191968865 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2a^2-a)\) | \(7\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
\((3a^2-a+1)\) | \(37\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(3\) | 3Cs.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 9 and 27.
Its isogeny class
259.1-A
consists of curves linked by isogenies of
degrees dividing 27.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.