Base field 4.4.2777.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 4 x^{2} + x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 1, -4, -1, 1]))
gp: K = nfinit(Polrev([2, 1, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 1, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1,0,0]),K([-2,-1,1,0]),K([-2,-1,1,0]),K([-77,-122,1,27]),K([327,512,-32,-103])])
gp: E = ellinit([Polrev([0,1,0,0]),Polrev([-2,-1,1,0]),Polrev([-2,-1,1,0]),Polrev([-77,-122,1,27]),Polrev([327,512,-32,-103])], K);
magma: E := EllipticCurve([K![0,1,0,0],K![-2,-1,1,0],K![-2,-1,1,0],K![-77,-122,1,27],K![327,512,-32,-103]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^3+5a^2-a-6)\) | = | \((-a)^{6}\cdot(-a^3+2a^2+2a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 704 \) | = | \(2^{6}\cdot11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((871a^3-565a^2-2604a-44)\) | = | \((-a)^{34}\cdot(-a^3+2a^2+2a-1)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 2078764171264 \) | = | \(2^{34}\cdot11^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{22702786953003143813}{7929856} a^{3} + \frac{30928910991628706537}{7929856} a^{2} - \frac{8873274696408279559}{3964928} a - \frac{19220593607745698479}{7929856} \) | ||
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: | $\left(-2 a^{3} + \frac{3}{4} a^{2} + 7 a + 6 : \frac{5}{8} a^{3} - \frac{7}{2} a - 1 : 1\right)$ |
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: | \( 53.856972788828019140987035063703612587 \) | ||
Tamagawa product: | \( 8 \) = \(2^{2}\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.04401460481171 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a)\) | \(2\) | \(4\) | \(I_{24}^{*}\) | Additive | \(-1\) | \(6\) | \(34\) | \(16\) |
\((-a^3+2a^2+2a-1)\) | \(11\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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, 4, 6, 8, 12 and 24.
Its isogeny class
704.3-q
consists of curves linked by isogenies of
degrees dividing 24.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.