Base field 3.3.961.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 10 x + 8 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([8, -10, -1, 1]))
gp: K = nfinit(Polrev([8, -10, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -10, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0]),K([-4,1/2,1/2]),K([1,0,0]),K([6274513813224967158312755214749080503789,-11880170148308365383903785310394241017191/2,-5171878668446132260474771433238959743711/2]),K([287620492634249855289661836746352099769467748294166719671064,-544580901779687961756431103089298691196301520328621842187519/2,-237076263554909853535311735052515444474061429406286932174419/2])])
gp: E = ellinit([Polrev([1,0,0]),Polrev([-4,1/2,1/2]),Polrev([1,0,0]),Polrev([6274513813224967158312755214749080503789,-11880170148308365383903785310394241017191/2,-5171878668446132260474771433238959743711/2]),Polrev([287620492634249855289661836746352099769467748294166719671064,-544580901779687961756431103089298691196301520328621842187519/2,-237076263554909853535311735052515444474061429406286932174419/2])], K);
magma: E := EllipticCurve([K![1,0,0],K![-4,1/2,1/2],K![1,0,0],K![6274513813224967158312755214749080503789,-11880170148308365383903785310394241017191/2,-5171878668446132260474771433238959743711/2],K![287620492634249855289661836746352099769467748294166719671064,-544580901779687961756431103089298691196301520328621842187519/2,-237076263554909853535311735052515444474061429406286932174419/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2)\) | = | \((-1/2a^2-1/2a+3)\cdot(-1/2a^2+1/2a+4)\cdot(a^2-11)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 8 \) | = | \(2\cdot2\cdot2\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-30311a^2+139005a+141172)\) | = | \((-1/2a^2-1/2a+3)\cdot(-1/2a^2+1/2a+4)^{6}\cdot(a^2-11)^{48}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 36028797018963968 \) | = | \(2\cdot2^{6}\cdot2^{48}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{41596750318948211}{281474976710656} a^{2} + \frac{62483684756063045}{140737488355328} a - \frac{15568191246577693}{35184372088832} \) | ||
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: | $\left(\frac{55794320654162058869}{4} a^{2} + 32040872629358785506 a - \frac{135379137093803308785}{4} : -\frac{55794320654162058869}{8} a^{2} - 16020436314679392753 a + \frac{135379137093803308781}{8} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 3.9375988015703234022248849925981285861 \) | ||
Tamagawa product: | \( 4 \) = \(1\cdot2\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.1431738456171906651620633849478437830 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-1/2a^2-1/2a+3)\) | \(2\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
\((-1/2a^2+1/2a+4)\) | \(2\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
\((a^2-11)\) | \(2\) | \(2\) | \(I_{48}\) | Non-split multiplicative | \(1\) | \(1\) | \(48\) | \(48\) |
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, 4, 6, 8, 12 and 24.
Its isogeny class
8.1-b
consists of curves linked by isogenies of
degrees dividing 24.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.