Base field 6.6.703493.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 5 x^{4} + 11 x^{3} + 2 x^{2} - 9 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -9, 2, 11, -5, -2, 1]))
gp: K = nfinit(Polrev([1, -9, 2, 11, -5, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 2, 11, -5, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,9,10,-11,-2,2]),K([2,-9,-1,7,0,-1]),K([2,8,0,-6,0,1]),K([6,-20,5,8,-1,-1]),K([-2,-1,16,-8,-4,2])])
gp: E = ellinit([Polrev([-3,9,10,-11,-2,2]),Polrev([2,-9,-1,7,0,-1]),Polrev([2,8,0,-6,0,1]),Polrev([6,-20,5,8,-1,-1]),Polrev([-2,-1,16,-8,-4,2])], K);
magma: E := EllipticCurve([K![-3,9,10,-11,-2,2],K![2,-9,-1,7,0,-1],K![2,8,0,-6,0,1],K![6,-20,5,8,-1,-1],K![-2,-1,16,-8,-4,2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((3a^5-2a^4-17a^3+10a^2+16a-3)\) | = | \((3a^5-2a^4-17a^3+10a^2+16a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 13 \) | = | \(13\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((6a^4+2a^3-21a^2-7a-10)\) | = | \((3a^5-2a^4-17a^3+10a^2+16a-3)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -62748517 \) | = | \(-13^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{33889457777}{62748517} a^{5} - \frac{83113788886}{62748517} a^{4} - \frac{102300181186}{62748517} a^{3} + \frac{371472408155}{62748517} a^{2} - \frac{227617587843}{62748517} a + \frac{15310089083}{62748517} \) | ||
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 | $\left(12 a^{5} - 8 a^{4} - 72 a^{3} + 39 a^{2} + 78 a - 10 : 40 a^{5} - 28 a^{4} - 236 a^{3} + 134 a^{2} + 247 a - 32 : 1\right)$ |
Height | \(0.0023491183090318897384186167089784011836\) |
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: | \( 0.0023491183090318897384186167089784011836 \) | ||
Period: | \( 17461.840809504202347106487193485287286 \) | ||
Tamagawa product: | \( 7 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.05407 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((3a^5-2a^4-17a^3+10a^2+16a-3)\) | \(13\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 13.1-b consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.