Base field 4.4.11661.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, 5, -4, -2, 1]))
gp: K = nfinit(Polrev([3, 5, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 5, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([2,-4,-1,1]),K([-3,-6,0,1]),K([-2,-1,1,0]),K([-42,30,18,-9]),K([107,-57,-44,19])])
gp: E = ellinit([Polrev([2,-4,-1,1]),Polrev([-3,-6,0,1]),Polrev([-2,-1,1,0]),Polrev([-42,30,18,-9]),Polrev([107,-57,-44,19])], K);
magma: E := EllipticCurve([K![2,-4,-1,1],K![-3,-6,0,1],K![-2,-1,1,0],K![-42,30,18,-9],K![107,-57,-44,19]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^3-a^2-5a)\) | = | \((-a)\cdot(a+1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 27 \) | = | \(3\cdot3^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-26a^3-25a^2+100a-57)\) | = | \((-a)^{15}\cdot(a+1)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 387420489 \) | = | \(3^{15}\cdot3^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{16988069332}{14348907} a^{3} + \frac{37528265996}{14348907} a^{2} + \frac{65539602622}{14348907} a - \frac{123694858235}{14348907} \) | ||
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 | $\left(-2 a^{3} + 11 a + 7 : 5 a^{3} - 4 a^{2} - 22 a - 6 : 1\right)$ |
Height | \(0.0071234152925676879924946239132997808055\) |
Torsion structure: | trivial |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(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: | \( 0.0071234152925676879924946239132997808055 \) | ||
Period: | \( 408.46516301754559657854036357247184625 \) | ||
Tamagawa product: | \( 30 \) = \(( 3 \cdot 5 )\cdot2\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 3.23337916276245 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a)\) | \(3\) | \(15\) | \(I_{15}\) | Split multiplicative | \(-1\) | \(1\) | \(15\) | \(15\) |
\((a+1)\) | \(3\) | \(2\) | \(III\) | Additive | \(1\) | \(2\) | \(3\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
27.1-e
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.