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([2,-3,-1,1]),K([4,-2,-2,1]),K([-2,-1,1,0]),K([-24086,21949,13289,-7800]),K([-1630220,1603430,935695,-559021])])
gp: E = ellinit([Polrev([2,-3,-1,1]),Polrev([4,-2,-2,1]),Polrev([-2,-1,1,0]),Polrev([-24086,21949,13289,-7800]),Polrev([-1630220,1603430,935695,-559021])], K);
magma: E := EllipticCurve([K![2,-3,-1,1],K![4,-2,-2,1],K![-2,-1,1,0],K![-24086,21949,13289,-7800],K![-1630220,1603430,935695,-559021]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-2a^2+8)\) | = | \((-a)^{6}\cdot(a^3-a^2-4a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 512 \) | = | \(2^{6}\cdot8\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-896a^3+768a^2+2688a+256)\) | = | \((-a)^{20}\cdot(a^3-a^2-4a+1)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -2199023255552 \) | = | \(-2^{20}\cdot8^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{414403249587808247093}{128} a^{3} - \frac{696044271483193936579}{128} a^{2} - \frac{592282302218182028565}{64} a + \frac{76217710069831417535}{8} \) | ||
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);
| |
Torsion generator: | $\left(-\frac{127}{4} a^{3} + 41 a^{2} + 94 a - 68 : \frac{181}{4} a^{3} - \frac{281}{4} a^{2} - \frac{987}{8} a + \frac{525}{4} : 1\right)$ |
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: | \( 1.7327258266057826810730197574109227860 \) | ||
Tamagawa product: | \( 4 \) = \(2^{2}\cdot1\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.61115746860280 \) | ||
Analytic order of Ш: | \( 49 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a)\) | \(2\) | \(4\) | \(I_{10}^{*}\) | Additive | \(1\) | \(6\) | \(20\) | \(2\) |
\((a^3-a^2-4a+1)\) | \(8\) | \(1\) | \(I_{7}\) | Non-split multiplicative | \(1\) | \(1\) | \(7\) | \(7\) |
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 |
\(7\) | 7B.6.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 7 and 14.
Its isogeny class
512.3-g
consists of curves linked by isogenies of
degrees dividing 14.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.