Base field 3.3.1957.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 9 x + 10 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([10, -9, -1, 1]))
gp: K = nfinit(Polrev([10, -9, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10, -9, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-6,0,1]),K([-6,1,1]),K([0,0,0]),K([-131023,95091,-17202]),K([-27013555,27777935,-6354008])])
gp: E = ellinit([Polrev([-6,0,1]),Polrev([-6,1,1]),Polrev([0,0,0]),Polrev([-131023,95091,-17202]),Polrev([-27013555,27777935,-6354008])], K);
magma: E := EllipticCurve([K![-6,0,1],K![-6,1,1],K![0,0,0],K![-131023,95091,-17202],K![-27013555,27777935,-6354008]]);
This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a-2)\) | = | \((2,a)^{4}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 16 \) | = | \(2^{4}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((8299a^2-6118a-61528)\) | = | \((2,a)^{44}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 17592186044416 \) | = | \(2^{44}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((-145a^2+671a-1582)\) | = | \((2,a)^{32}\) |
Minimal discriminant norm: | \( 4294967296 \) | = | \(2^{32}\) |
j-invariant: | \( \frac{1208201331713038647}{1048576} a^{2} - \frac{4882016462354803125}{1048576} a + \frac{3972767044836889003}{1048576} \) | ||
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: | \(2\) | |
Generators | $\left(-\frac{15198372}{24649} a^{2} - \frac{153288}{24649} a + \frac{129225633}{24649} : \frac{149018426588}{3869893} a^{2} + \frac{18829071819}{3869893} a - \frac{1318017318621}{3869893} : 1\right)$ | $\left(-\frac{728985}{1444} a^{2} + \frac{11907}{1444} a + \frac{6141583}{1444} : -\frac{1666184591}{54872} a^{2} - \frac{222272819}{54872} a + \frac{14771300301}{54872} : 1\right)$ |
Heights | \(4.8725588887586626182133103359233414024\) | \(3.6716429585045910472885789364813449628\) |
Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
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 generators: | $\left(53 a^{2} - 141 a - 41 : -15 a^{2} + 238 a - 563 : 1\right)$ | $\left(-\frac{5}{2} a^{2} + \frac{293}{4} a - \frac{389}{2} : \frac{525}{8} a^{2} - \frac{889}{8} a - \frac{919}{4} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 2 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(2\) | ||
Regulator: | \( 16.206663733510544239524957340459925355 \) | ||
Period: | \( 5.3955965256126612839868792917594925785 \) | ||
Tamagawa product: | \( 4 \) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 4.4475433805892265129769494578830631605 \) | ||
Analytic order of Ш: | \( 1 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
Primes of good reduction for the curve but which divide the
discriminant of the model above (if any) are included.
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(4\) | \(I_{24}^{*}\) | Additive | \(-1\) | \(4\) | \(32\) | \(20\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2Cs |
\(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
16.3-b
consists of curves linked by isogenies of
degrees dividing 20.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.