Base field 4.4.18625.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 14 x^{2} + 9 x + 41 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([41, 9, -14, -1, 1]))
gp: K = nfinit(Polrev([41, 9, -14, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 9, -14, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-35/6,-1/3,1,1/6]),K([-41/6,-4/3,1,1/6]),K([-35/6,-1/3,1,1/6]),K([137,-298,-9,33]),K([11145/2,-3184,-575,675/2])])
gp: E = ellinit([Polrev([-35/6,-1/3,1,1/6]),Polrev([-41/6,-4/3,1,1/6]),Polrev([-35/6,-1/3,1,1/6]),Polrev([137,-298,-9,33]),Polrev([11145/2,-3184,-575,675/2])], K);
magma: E := EllipticCurve([K![-35/6,-1/3,1,1/6],K![-41/6,-4/3,1,1/6],K![-35/6,-1/3,1,1/6],K![137,-298,-9,33],K![11145/2,-3184,-575,675/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-1/3a^3-a^2+8/3a+20/3)\) | = | \((-a+3)\cdot(1/2a^3+a^2-5a-21/2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 20 \) | = | \(4\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((25/3a^3+50a^2-275/3a-1400/3)\) | = | \((-a+3)^{2}\cdot(1/2a^3+a^2-5a-21/2)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 156250000 \) | = | \(4^{2}\cdot5^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{4461364253}{1500} a^{3} + \frac{3377251529}{500} a^{2} - \frac{5864955929}{300} a - \frac{13968231781}{375} \) | ||
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 | $\left(\frac{1442}{3} a^{3} - 1273 a^{2} - \frac{13900}{3} a + \frac{35870}{3} : -\frac{168121}{3} a^{3} + 148396 a^{2} + \frac{1620113}{3} a - \frac{4182994}{3} : 1\right)$ |
Height | \(1.1573605113322744832563237188164128100\) |
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: | $\left(-\frac{19}{12} a^{3} - a^{2} + \frac{179}{12} a + \frac{23}{3} : \frac{43}{24} a^{3} + \frac{53}{8} a^{2} - \frac{311}{24} a - \frac{1211}{24} : 1\right)$ |
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: | \( 1.1573605113322744832563237188164128100 \) | ||
Period: | \( 131.75493511223892958486148949421220989 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 4.46938124685478 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a+3)\) | \(4\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((1/2a^3+a^2-5a-21/2)\) | \(5\) | \(2\) | \(I_{10}\) | 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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
20.1-c
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.