Base field 6.6.592661.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 5 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 5, 4, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, -2, 5, 4, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 5, 4, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,5,4,-9,-1,2]),K([3,0,-1,0,0,0]),K([-4,5,5,-9,-1,2]),K([25,28,-126,46,33,-15]),K([98,117,-590,190,155,-62])])
gp: E = ellinit([Polrev([-3,5,4,-9,-1,2]),Polrev([3,0,-1,0,0,0]),Polrev([-4,5,5,-9,-1,2]),Polrev([25,28,-126,46,33,-15]),Polrev([98,117,-590,190,155,-62])], K);
magma: E := EllipticCurve([K![-3,5,4,-9,-1,2],K![3,0,-1,0,0,0],K![-4,5,5,-9,-1,2],K![25,28,-126,46,33,-15],K![98,117,-590,190,155,-62]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^5+2a^4+5a^3-8a^2-4a+3)\) | = | \((-a^5+2a^4+5a^3-8a^2-4a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 73 \) | = | \(73\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-54a^5+301a^4-5a^3-1177a^2+610a-263)\) | = | \((-a^5+2a^4+5a^3-8a^2-4a+3)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 58871586708267913 \) | = | \(73^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{819409026299375534395416}{58871586708267913} a^{5} - \frac{170251602571138272825926}{58871586708267913} a^{4} + \frac{3968508698135623357582134}{58871586708267913} a^{3} + \frac{1769334738717973761105523}{58871586708267913} a^{2} - \frac{1771210861015578326012193}{58871586708267913} a - \frac{587323218030274014006283}{58871586708267913} \) | ||
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 | $\left(-9 a^{5} + 7 a^{4} + 38 a^{3} - 27 a^{2} - 15 a + 17 : -69 a^{5} + 34 a^{4} + 307 a^{3} - 124 a^{2} - 161 a + 97 : 1\right)$ |
Height | \(0.13629071148651834659514730526851115249\) |
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);
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Torsion generator: | $\left(a^{5} + a^{4} - 6 a^{3} - 4 a^{2} + 7 a - 1 : 4 a^{5} - a^{4} - 18 a^{3} + 2 a^{2} + 10 a - 2 : 1\right)$ |
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: | \( 0.13629071148651834659514730526851115249 \) | ||
Period: | \( 122.21723548175440316502011731765875640 \) | ||
Tamagawa product: | \( 9 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.62889 \) | ||
Analytic order of Ш: | \( 9 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^5+2a^4+5a^3-8a^2-4a+3)\) | \(73\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
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 |
\(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
73.2-b
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.