Base field 6.6.485125.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -5, 2, 8, -4, -2, 1]))
gp: K = nfinit(Polrev([1, -5, 2, 8, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,1,1,0,0,0]),K([-3,6,10,-8,-3,2]),K([4,-3,-7,4,2,-1]),K([-40,-71,5,42,2,-6]),K([153,157,-91,-89,11,11])])
gp: E = ellinit([Polrev([-1,1,1,0,0,0]),Polrev([-3,6,10,-8,-3,2]),Polrev([4,-3,-7,4,2,-1]),Polrev([-40,-71,5,42,2,-6]),Polrev([153,157,-91,-89,11,11])], K);
magma: E := EllipticCurve([K![-1,1,1,0,0,0],K![-3,6,10,-8,-3,2],K![4,-3,-7,4,2,-1],K![-40,-71,5,42,2,-6],K![153,157,-91,-89,11,11]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^5+3a^4+10a^3-12a^2-11a+6)\) | = | \((-2a^5+3a^4+10a^3-12a^2-11a+6)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 49 \) | = | \(49\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-5a^5+6a^4+22a^3-20a^2-17a+8)\) | = | \((-2a^5+3a^4+10a^3-12a^2-11a+6)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -2401 \) | = | \(-49^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{44682954498275}{49} a^{5} + \frac{20731677864676}{49} a^{4} + \frac{214541279828608}{49} a^{3} - \frac{20858262562702}{49} a^{2} - \frac{124233907036096}{49} a + \frac{28389749219352}{49} \) | ||
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: | \(0\) |
| 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: | $\left(-6 a^{5} + \frac{35}{4} a^{4} + \frac{55}{2} a^{3} - \frac{127}{4} a^{2} - \frac{55}{2} a + \frac{51}{4} : \frac{5}{8} a^{5} - \frac{13}{8} a^{3} + \frac{3}{4} a^{2} + \frac{3}{2} a - \frac{1}{4} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 2329.3641775987155325503979334636668359 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.67217 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^5+3a^4+10a^3-12a^2-11a+6)\) | \(49\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
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
49.1-a
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.