Base field 6.6.1387029.1
Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 2 x^{4} + 9 x^{3} - x^{2} - 4 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -4, -1, 9, -2, -3, 1]))
gp: K = nfinit(Polrev([1, -4, -1, 9, -2, -3, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -1, 9, -2, -3, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,3,-2,-2,1,0]),K([-2,-1,1,0,0,0]),K([4,-4,-13,6,5,-2]),K([-1143,-1196,979,455,-232,1]),K([-15855,-16196,13092,6206,-3103,0])])
gp: E = ellinit([Polrev([1,3,-2,-2,1,0]),Polrev([-2,-1,1,0,0,0]),Polrev([4,-4,-13,6,5,-2]),Polrev([-1143,-1196,979,455,-232,1]),Polrev([-15855,-16196,13092,6206,-3103,0])], K);
magma: E := EllipticCurve([K![1,3,-2,-2,1,0],K![-2,-1,1,0,0,0],K![4,-4,-13,6,5,-2],K![-1143,-1196,979,455,-232,1],K![-15855,-16196,13092,6206,-3103,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-2a^5+5a^4+6a^3-14a^2-3a+4)\) | = | \((a^5-3a^4-2a^3+8a^2+a-2)\cdot(a^2-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 21 \) | = | \(3\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-16a^4+32a^3+57a^2-73a-28)\) | = | \((a^5-3a^4-2a^3+8a^2+a-2)^{4}\cdot(a^2-2)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 466948881 \) | = | \(3^{4}\cdot7^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{2905242808362156164576920441}{21609} a^{4} - \frac{5810485616724312329153840882}{21609} a^{3} - \frac{12198217539758009162689930484}{21609} a^{2} + \frac{719212397529531682250802425}{1029} a + \frac{14621896551411174957571933555}{21609} \) | ||
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 | $\left(-14 a^{5} + 23 a^{4} + 86 a^{3} - 97 a^{2} - 134 a + 91 : -36 a^{5} + 214 a^{4} - 309 a^{3} - 326 a^{2} + 952 a - 435 : 1\right)$ |
Height | \(1.1731957583480136207825413586508381139\) |
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: | $\left(-\frac{9}{2} a^{4} + 9 a^{3} + \frac{69}{4} a^{2} - \frac{87}{4} a - \frac{33}{2} : a^{5} - \frac{1}{4} a^{4} - \frac{15}{2} a^{3} - \frac{29}{8} a^{2} + \frac{115}{8} a + \frac{101}{8} : 1\right)$ |
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: | \( 1.1731957583480136207825413586508381139 \) | ||
Period: | \( 1.3086722316626959187897553071279438812 \) | ||
Tamagawa product: | \( 8 \) = \(2^{2}\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 4.00479 \) | ||
Analytic order of Ш: | \( 256 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^5-3a^4-2a^3+8a^2+a-2)\) | \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
\((a^2-2)\) | \(7\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
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, 4 and 8.
Its isogeny class
21.1-b
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
3.3.257.1 | 3.3.257.1-21.1-b6 |
3.3.257.1 | a curve with conductor norm 441 (not in the database) |