Base field 4.4.9909.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - 3 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -3, -6, 0, 1]))
gp: K = nfinit(Polrev([3, -3, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -3, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0,0]),K([0,1,0,0]),K([-2,0,1,0]),K([-34158,62447,16600,-13750]),K([3262240,-5963840,-1585569,1313070])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([0,1,0,0]),Polrev([-2,0,1,0]),Polrev([-34158,62447,16600,-13750]),Polrev([3262240,-5963840,-1585569,1313070])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![0,1,0,0],K![-2,0,1,0],K![-34158,62447,16600,-13750],K![3262240,-5963840,-1585569,1313070]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a-3)\) | = | \((-a^3+a^2+4a)\cdot(a^3-a^2-4a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 21 \) | = | \(3\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((-2487a^3+2531a^2+9684a-1314)\) | = | \((-a^3+a^2+4a)^{2}\cdot(a^3-a^2-4a+1)^{16}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 299096375126409 \) | = | \(3^{2}\cdot7^{16}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( \frac{3075730896093395126238965002}{99698791708803} a^{3} - \frac{3714007662409537173380188813}{99698791708803} a^{2} - \frac{4656548508714567651606714255}{33232930569601} a + \frac{2547146210989604914629326570}{33232930569601} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
Sato-Tate group: | $\mathrm{SU}(2)$ |
Mordell-Weil group
Rank: | \(1\) | |
Generator | $\left(\frac{35588434}{1113025} a^{3} - \frac{41646111}{1113025} a^{2} - \frac{32178412}{222605} a + \frac{88379063}{1113025} : -\frac{20256966794}{1174241375} a^{3} + \frac{19955954476}{1174241375} a^{2} + \frac{17108767582}{234848275} a - \frac{45430592508}{1174241375} : 1\right)$ | |
Height | \(4.2563373253223029300708455110830687319\) | |
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);
| ||
Torsion generators: | $\left(32 a^{3} - 38 a^{2} - 145 a + \frac{319}{4} : -16 a^{3} + \frac{37}{2} a^{2} + \frac{145}{2} a - \frac{311}{8} : 1\right)$ | $\left(32 a^{3} - 39 a^{2} - 146 a + 80 : -16 a^{3} + 19 a^{2} + 73 a - 39 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
|
|||
Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 4.2563373253223029300708455110830687319 \) | ||
Period: | \( 87.105030326823751242452985106402193751 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 3.72446898122785 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^3+a^2+4a)\) | \(3\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((a^3-a^2-4a+1)\) | \(7\) | \(2\) | \(I_{16}\) | Non-split multiplicative | \(1\) | \(1\) | \(16\) | \(16\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
21.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.