Base field 6.6.434581.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 5 x^{3} + 4 x^{2} - 2 x - 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 4, 5, -4, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 4*x^4 + 5*x^3 + 4*x^2 - 2*x - 1)
gp (2.8): K = nfinit(a^6 - 2*a^5 - 4*a^4 + 5*a^3 + 4*a^2 - 2*a - 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([3*a^5 - 6*a^4 - 10*a^3 + 12*a^2 + 4*a - 2, -a^5 + 6*a^3 + 4*a^2 - 5*a - 3, a^5 - 2*a^4 - 3*a^3 + 4*a^2 - 1, -297*a^5 + 798*a^4 + 604*a^3 - 1840*a^2 + 208*a + 366, -3433*a^5 + 9442*a^4 + 6587*a^3 - 22010*a^2 + 3003*a + 4476]),K);
sage: E = EllipticCurve(K, [3*a^5 - 6*a^4 - 10*a^3 + 12*a^2 + 4*a - 2, -a^5 + 6*a^3 + 4*a^2 - 5*a - 3, a^5 - 2*a^4 - 3*a^3 + 4*a^2 - 1, -297*a^5 + 798*a^4 + 604*a^3 - 1840*a^2 + 208*a + 366, -3433*a^5 + 9442*a^4 + 6587*a^3 - 22010*a^2 + 3003*a + 4476])
gp (2.8): E = ellinit([3*a^5 - 6*a^4 - 10*a^3 + 12*a^2 + 4*a - 2, -a^5 + 6*a^3 + 4*a^2 - 5*a - 3, a^5 - 2*a^4 - 3*a^3 + 4*a^2 - 1, -297*a^5 + 798*a^4 + 604*a^3 - 1840*a^2 + 208*a + 366, -3433*a^5 + 9442*a^4 + 6587*a^3 - 22010*a^2 + 3003*a + 4476],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((71,2 a^{5} - 6 a^{4} - 4 a^{3} + 17 a^{2} - a - 6)\) | = | \( \left(2 a^{5} - 6 a^{4} - 4 a^{3} + 17 a^{2} - a - 6\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 71 \) | = | \( 71 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((357911,a^{4} - 2 a^{3} - 3 a^{2} + 3 a + 261300,a^{5} - 3 a^{4} - 2 a^{3} + 8 a^{2} + 82424,a + 279766,a^{5} - 2 a^{4} - 3 a^{3} + 3 a^{2} + a + 29894,a^{5} - 2 a^{4} - 3 a^{3} + 4 a^{2} + 144496)\) | = | \( \left(2 a^{5} - 6 a^{4} - 4 a^{3} + 17 a^{2} - a - 6\right)^{3} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 357911 \) | = | \( 71^{3} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -\frac{25415776248234932727}{357911} a^{5} + \frac{33644549177505288315}{357911} a^{4} + \frac{124407707776619205840}{357911} a^{3} - \frac{42935695592915487066}{357911} a^{2} - \frac{130674592083511357736}{357911} a - \frac{37564287471550457576}{357911} \) | ||
| magma: jInvariant(E);
sage: E.j_invariant()
gp (2.8): E.j
| ||||
| \( \text{End} (E) \) | = | \(\Z\) | (no Complex Multiplication ) | |
| magma: HasComplexMultiplication(E);
sage: E.has_cm(), E.cm_discriminant()
| ||||
| \( \text{ST} (E) \) | = | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
Rank not available.magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()
Regulator: not available
magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())
Torsion subgroup
| Structure: | Trivial |
|---|---|
| magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[2]
| |
| magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp (2.8): elltors(E)[1]
| |
Local data at primes of bad reduction
magma: LocalInformation(E);
sage: E.local_data()
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(2 a^{5} - 6 a^{4} - 4 a^{3} + 17 a^{2} - a - 6\right) \) | \(71\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
71.1-b
consists of curves linked by isogenies of
degree3.