Base field 4.4.2624.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 2, -3, -2, 1]))
gp: K = nfinit(Polrev([1, 2, -3, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -3, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-1,1,0]),K([-2,-2,1,0]),K([2,-1,-2,1]),K([-13,53,-62,15]),K([-6,-211,335,-93])])
gp: E = ellinit([Polrev([-1,-1,1,0]),Polrev([-2,-2,1,0]),Polrev([2,-1,-2,1]),Polrev([-13,53,-62,15]),Polrev([-6,-211,335,-93])], K);
magma: E := EllipticCurve([K![-1,-1,1,0],K![-2,-2,1,0],K![2,-1,-2,1],K![-13,53,-62,15],K![-6,-211,335,-93]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^3-3a^2+4)\) | = | \((-a^2+a+2)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 49 \) | = | \(7^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-9a^3+24a^2+19a-13)\) | = | \((-a^2+a+2)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 117649 \) | = | \(7^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( 427641453408640 a^{3} + 331260963910720 a^{2} - 363800343820288 a - 154126073580480 \) | ||
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: | \(1\) |
Generator | $\left(-7 a^{3} + 9 a^{2} + 25 a + 12 : -45 a^{3} + 51 a^{2} + 183 a + 56 : 1\right)$ |
Height | \(0.11509089569110000142749127678736681871\) |
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: | $\left(-4 a^{3} + \frac{19}{2} a^{2} + 7 a - 6 : -\frac{3}{4} a^{3} + \frac{9}{2} a - \frac{5}{4} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.11509089569110000142749127678736681871 \) | ||
Period: | \( 312.55080983089286734198401157859468437 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.40446098336357 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^2+a+2)\) | \(7\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2B |
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
49.3-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.