Base field 3.3.733.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 7 x + 8 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([8, -7, -1, 1]))
gp: K = nfinit(Polrev([8, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1,0]),K([-5,1,1]),K([-5,1,1]),K([-225,1,33]),K([1450,-51,-218])])
gp: E = ellinit([Polrev([0,1,0]),Polrev([-5,1,1]),Polrev([-5,1,1]),Polrev([-225,1,33]),Polrev([1450,-51,-218])], K);
magma: E := EllipticCurve([K![0,1,0],K![-5,1,1],K![-5,1,1],K![-225,1,33],K![1450,-51,-218]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-2a^2-2a+19)\) | = | \((-a+3)^{2}\cdot(a^2+2a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 175 \) | = | \(5^{2}\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-4057a^2-3034a+15945)\) | = | \((-a+3)^{6}\cdot(a^2+2a-3)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -630525109375 \) | = | \(-5^{6}\cdot7^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{16146410084579303}{40353607} a^{2} - \frac{24513786018922996}{40353607} a + \frac{51294295430574624}{40353607} \) | ||
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: | trivial |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(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: | \( 4.7613092227999599195686230608995065758 \) | ||
Tamagawa product: | \( 9 \) = \(1\cdot3^{2}\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 1.5827667189652448317138194236475424018 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a+3)\) | \(5\) | \(1\) | \(I_0^{*}\) | Additive | \(1\) | \(2\) | \(6\) | \(0\) |
\((a^2+2a-3)\) | \(7\) | \(9\) | \(I_{9}\) | 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 |
---|---|
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
175.2-a
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.