Base field 4.4.16400.1
Generator \(a\), with minimal polynomial \( x^{4} - 13 x^{2} + 41 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([41, 0, -13, 0, 1]))
gp: K = nfinit(Polrev([41, 0, -13, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 0, -13, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,-6,0,1]),K([6,-1,-1,0]),K([-6,0,1,0]),K([-216,102,24,-12]),K([-1386,633,154,-73])])
gp: E = ellinit([Polrev([0,-6,0,1]),Polrev([6,-1,-1,0]),Polrev([-6,0,1,0]),Polrev([-216,102,24,-12]),Polrev([-1386,633,154,-73])], K);
magma: E := EllipticCurve([K![0,-6,0,1],K![6,-1,-1,0],K![-6,0,1,0],K![-216,102,24,-12],K![-1386,633,154,-73]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^2+a-4)\) | = | \((a^2+a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 19 \) | = | \(19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-a^2-a+4)\) | = | \((a^2+a-4)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -19 \) | = | \(-19\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{24005877695}{19} a^{3} - \frac{66262595100}{19} a^{2} + \frac{129187416855}{19} a + \frac{356600683675}{19} \) | ||
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 | $\left(-\frac{8}{5} a^{3} + \frac{99}{25} a^{2} + \frac{266}{25} a - \frac{639}{25} : \frac{23}{125} a^{3} - \frac{132}{125} a^{2} - \frac{339}{125} a + \frac{1284}{125} : 1\right)$ |
Height | \(1.8197258685595835488627398475984425741\) |
Torsion structure: | \(\Z/3\Z\) |
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: | $\left(a^{3} - 2 a^{2} - 10 a + 22 : 8 a^{3} - 20 a^{2} - 58 a + 143 : 1\right)$ |
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: | \( 1.8197258685595835488627398475984425741 \) | ||
Period: | \( 517.43482745405246445223821110320423552 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(3\) | ||
Leading coefficient: | \( 3.26781290541210 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^2+a-4)\) | \(19\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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.1.1 |
\(5\) | 5B[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5 and 15.
Its isogeny class
19.1-b
consists of curves linked by isogenies of
degrees dividing 15.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.