Base field 4.4.4525.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + 3 x + 9 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([9, 3, -7, -1, 1]))
gp: K = nfinit(Polrev([9, 3, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 3, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,0,1,0]),K([3,-4/3,-4/3,1/3]),K([-3,-7/3,2/3,1/3]),K([-623,-1894/3,131/3,295/3]),K([7921,24299/3,-1723/3,-3827/3])])
gp: E = ellinit([Polrev([-3,0,1,0]),Polrev([3,-4/3,-4/3,1/3]),Polrev([-3,-7/3,2/3,1/3]),Polrev([-623,-1894/3,131/3,295/3]),Polrev([7921,24299/3,-1723/3,-3827/3])], K);
magma: E := EllipticCurve([K![-3,0,1,0],K![3,-4/3,-4/3,1/3],K![-3,-7/3,2/3,1/3],K![-623,-1894/3,131/3,295/3],K![7921,24299/3,-1723/3,-3827/3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^2+6)\) | = | \((-a^2+a+5)\cdot(-a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 45 \) | = | \(5\cdot9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((108a^3-65a^2-390a-63)\) | = | \((-a^2+a+5)\cdot(-a)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 17433922005 \) | = | \(5\cdot9^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{24210679621695774874}{885735} a^{3} + \frac{14857696483628142620}{177147} a^{2} + \frac{15815131470306049663}{885735} a - \frac{35114779889903532878}{295245} \) | ||
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: | \(\Z/2\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(-\frac{23}{12} a^{3} + \frac{5}{12} a^{2} + \frac{137}{12} a + 5 : -\frac{31}{24} a^{3} + \frac{1}{6} a^{2} + \frac{89}{12} a + \frac{9}{4} : 1\right)$ |
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: | \( 63.137482375433209382500214893679028566 \) | ||
Tamagawa product: | \( 2 \) = \(1\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.87718884233773 \) | ||
Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^2+a+5)\) | \(5\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
\((-a)\) | \(9\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
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 |
\(5\) | 5B.4.1[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
45.3-d
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.