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([-6*a^5 + a^4 + 44*a^3 + 23*a^2 - 28*a - 9, 3*a^5 - a^4 - 21*a^3 - 10*a^2 + 12*a + 6, a^5 - 7*a^3 - 5*a^2 + 3*a + 2, 301*a^5 - 10*a^4 - 2223*a^3 - 1411*a^2 + 1326*a + 389, 2885*a^5 - 180*a^4 - 21196*a^3 - 12957*a^2 + 12811*a + 3947]),K);
sage: E = EllipticCurve(K, [-6*a^5 + a^4 + 44*a^3 + 23*a^2 - 28*a - 9, 3*a^5 - a^4 - 21*a^3 - 10*a^2 + 12*a + 6, a^5 - 7*a^3 - 5*a^2 + 3*a + 2, 301*a^5 - 10*a^4 - 2223*a^3 - 1411*a^2 + 1326*a + 389, 2885*a^5 - 180*a^4 - 21196*a^3 - 12957*a^2 + 12811*a + 3947])
gp (2.8): E = ellinit([-6*a^5 + a^4 + 44*a^3 + 23*a^2 - 28*a - 9, 3*a^5 - a^4 - 21*a^3 - 10*a^2 + 12*a + 6, a^5 - 7*a^3 - 5*a^2 + 3*a + 2, 301*a^5 - 10*a^4 - 2223*a^3 - 1411*a^2 + 1326*a + 389, 2885*a^5 - 180*a^4 - 21196*a^3 - 12957*a^2 + 12811*a + 3947],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((71,8 a^{5} - 2 a^{4} - 58 a^{3} - 27 a^{2} + 38 a + 10)\) | = | \( \left(-a^{5} + a^{4} + 7 a^{3} - 2 a^{2} - 7 a\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 71 \) | = | \( 71 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((128100283921,-2 a^{5} + 15 a^{3} + 10 a^{2} - 10 a + 32213077914,3 a^{5} - a^{4} - 21 a^{3} - 9 a^{2} + 12 a + 78493378987,-4 a^{5} + a^{4} + 29 a^{3} + 13 a^{2} - 19 a + 10141833408,a + 85380698657,-2 a^{5} + a^{4} + 14 a^{3} + 4 a^{2} - 10 a + 53212805057)\) | = | \( \left(-a^{5} + a^{4} + 7 a^{3} - 2 a^{2} - 7 a\right)^{6} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 128100283921 \) | = | \( 71^{6} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -\frac{12188349191966910976545591421520911263}{128100283921} a^{5} + \frac{2795083691338226342388959981793334127}{128100283921} a^{4} + \frac{87472547643214485589591447553049289390}{128100283921} a^{3} + \frac{43036275357468336314681380908411365576}{128100283921} a^{2} - \frac{734527173722240035758902620347146720}{1804229351} a - \frac{15815145011644535445549863476994175471}{128100283921} \) | ||
| 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: | \(\Z/2\Z\) |
|---|---|
| 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: | $\left(\frac{29}{4} a^{5} - 5 a^{4} - \frac{187}{4} a^{3} - \frac{23}{4} a^{2} + \frac{67}{4} a + \frac{1}{4} : -33 a^{5} + \frac{87}{8} a^{4} + \frac{1861}{8} a^{3} + \frac{777}{8} a^{2} - \frac{1095}{8} a - \frac{333}{8} : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-a^{5} + a^{4} + 7 a^{3} - 2 a^{2} - 7 a\right) \) | \(71\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
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.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
71.3-c
consists of curves linked by isogenies of
degrees dividing 6.