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([-1,-4,0,1]),K([-4,3,2,-1]),K([-2,0,1,0]),K([5,-9,-7,4]),K([-3,1,2,-1])])
gp: E = ellinit([Polrev([-1,-4,0,1]),Polrev([-4,3,2,-1]),Polrev([-2,0,1,0]),Polrev([5,-9,-7,4]),Polrev([-3,1,2,-1])], K);
magma: E := EllipticCurve([K![-1,-4,0,1],K![-4,3,2,-1],K![-2,0,1,0],K![5,-9,-7,4],K![-3,1,2,-1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^3+3a)\) | = | \((-a)\cdot(-a^3+2a^2+2a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 22 \) | = | \(2\cdot11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((18a^3-19a^2-56a+12)\) | = | \((-a)^{10}\cdot(-a^3+2a^2+2a-1)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 123904 \) | = | \(2^{10}\cdot11^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{189687996477}{123904} a^{3} + \frac{252879329025}{123904} a^{2} - \frac{75995630703}{61952} a - \frac{153356764887}{123904} \) | ||
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: | \(\Z/2\Z\oplus\Z/10\Z\) | |
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 generators: | $\left(\frac{3}{4} a^{3} - 2 a^{2} - \frac{3}{4} a + \frac{5}{2} : -a^{3} + \frac{9}{8} a^{2} + \frac{19}{8} a + \frac{1}{4} : 1\right)$ | $\left(a^{3} - 2 a^{2} - 2 a + 2 : -1 : 1\right)$ |
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: | \( 1697.8947969928443846920318669362486947 \) | ||
Tamagawa product: | \( 20 \) = \(( 2 \cdot 5 )\cdot2\) | ||
Torsion order: | \(20\) | ||
Leading coefficient: | \( 1.61098999014251 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a)\) | \(2\) | \(10\) | \(I_{10}\) | Split multiplicative | \(-1\) | \(1\) | \(10\) | \(10\) |
\((-a^3+2a^2+2a-1)\) | \(11\) | \(2\) | \(I_{2}\) | 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 |
---|---|
\(2\) | 2Cs |
\(5\) | 5B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
22.1-b
consists of curves linked by isogenies of
degrees dividing 20.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.