Base field 3.3.257.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 4 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -4, -1, 1]))
gp: K = nfinit(Polrev([3, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,1,1]),K([1,0,0]),K([-3,1,1]),K([340,-280,-230]),K([4812,-4360,-3614])])
gp: E = ellinit([Polrev([-3,1,1]),Polrev([1,0,0]),Polrev([-3,1,1]),Polrev([340,-280,-230]),Polrev([4812,-4360,-3614])], K);
magma: E := EllipticCurve([K![-3,1,1],K![1,0,0],K![-3,1,1],K![340,-280,-230],K![4812,-4360,-3614]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^2+a-6)\) | = | \((a^2-3)\cdot(a^2-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 21 \) | = | \(3\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((16a^2+9a-54)\) | = | \((a^2-3)^{2}\cdot(a^2-2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 21609 \) | = | \(3^{2}\cdot7^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{2905242808362156164576920441}{21609} a^{2} + \frac{3482489114671540668959169161}{21609} a - \frac{3964052911380531038654086531}{21609} \) | ||
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: | \(\Z/2\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
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Torsion generator: | $\left(-\frac{9}{2} a^{2} - \frac{19}{4} a + 9 : \frac{13}{2} a^{2} + \frac{69}{8} a - \frac{45}{8} : 1\right)$ |
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: | \( 1.1439721288836961515590511042799053500 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.1417443329449322509612028816234440288 \) | ||
Analytic order of Ш: | \( 16 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^2-3)\) | \(3\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((a^2-2)\) | \(7\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
21.1-b
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.