Base field 6.6.300125.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 7, 2, -7, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 7*x^4 + 2*x^3 + 7*x^2 - 2*x - 1)
gp (2.8): K = nfinit(a^6 - a^5 - 7*a^4 + 2*a^3 + 7*a^2 - 2*a - 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([-4*a^5 + a^4 + 29*a^3 + 14*a^2 - 20*a - 7, a^5 - a^4 - 6*a^3 + 2*a, -4*a^5 + a^4 + 29*a^3 + 13*a^2 - 19*a - 5, 6*a^5 - 19*a^4 - 16*a^3 + 68*a^2 - 19*a - 13, -302*a^5 + 643*a^4 + 1380*a^3 - 2136*a^2 + 291*a + 268]),K);
sage: E = EllipticCurve(K, [-4*a^5 + a^4 + 29*a^3 + 14*a^2 - 20*a - 7, a^5 - a^4 - 6*a^3 + 2*a, -4*a^5 + a^4 + 29*a^3 + 13*a^2 - 19*a - 5, 6*a^5 - 19*a^4 - 16*a^3 + 68*a^2 - 19*a - 13, -302*a^5 + 643*a^4 + 1380*a^3 - 2136*a^2 + 291*a + 268])
gp (2.8): E = ellinit([-4*a^5 + a^4 + 29*a^3 + 14*a^2 - 20*a - 7, a^5 - a^4 - 6*a^3 + 2*a, -4*a^5 + a^4 + 29*a^3 + 13*a^2 - 19*a - 5, 6*a^5 - 19*a^4 - 16*a^3 + 68*a^2 - 19*a - 13, -302*a^5 + 643*a^4 + 1380*a^3 - 2136*a^2 + 291*a + 268],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((71,-8 a^{5} + a^{4} + 58 a^{3} + 34 a^{2} - 34 a - 16)\) | = | \( \left(4 a^{5} - a^{4} - 29 a^{3} - 13 a^{2} + 18 a + 3\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 71 \) | = | \( 71 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((5041,-2 a^{5} + 15 a^{3} + 10 a^{2} - 10 a + 1766,3 a^{5} - a^{4} - 21 a^{3} - 9 a^{2} + 12 a + 837,-4 a^{5} + a^{4} + 29 a^{3} + 13 a^{2} - 19 a + 3253,a + 1856,-2 a^{5} + a^{4} + 14 a^{3} + 4 a^{2} - 10 a + 2939)\) | = | \( \left(4 a^{5} - a^{4} - 29 a^{3} - 13 a^{2} + 18 a + 3\right)^{2} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 5041 \) | = | \( 71^{2} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -\frac{756505033288550895}{5041} a^{5} + \frac{964582520701492616}{5041} a^{4} + \frac{5030225884885221547}{5041} a^{3} - \frac{2896578761117562289}{5041} a^{2} - \frac{4498828390434099007}{5041} a + \frac{2750417335307539161}{5041} \) | ||
| 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: | \(\Z/6\Z\) |
|---|---|
| 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]
| |
| Generator: | $\left(10 a^{5} - 17 a^{4} - 52 a^{3} + 48 a^{2} + 2 a - 3 : -75 a^{5} + 161 a^{4} + 346 a^{3} - 555 a^{2} + 82 a + 71 : 1\right)$ |
| magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[3]
| |
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(4 a^{5} - a^{4} - 29 a^{3} - 13 a^{2} + 18 a + 3\right) \) | \(71\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
71.1-c
consists of curves linked by isogenies of
degrees dividing 6.