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 \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 4*x^4 + 8*x^3 + 2*x^2 - 5*x + 1)
gp (2.8): K = nfinit(a^6 - 2*a^5 - 4*a^4 + 8*a^3 + 2*a^2 - 5*a + 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a^2 + a - 1, 2*a^5 - 3*a^4 - 8*a^3 + 10*a^2 + 6*a - 3, -a^5 + 2*a^4 + 4*a^3 - 7*a^2 - 3*a + 4, -6*a^5 + 2*a^4 + 42*a^3 + 5*a^2 - 71*a - 40, 11*a^5 + 11*a^4 - 89*a^3 - 91*a^2 + 157*a + 153]),K);
sage: E = EllipticCurve(K, [a^2 + a - 1, 2*a^5 - 3*a^4 - 8*a^3 + 10*a^2 + 6*a - 3, -a^5 + 2*a^4 + 4*a^3 - 7*a^2 - 3*a + 4, -6*a^5 + 2*a^4 + 42*a^3 + 5*a^2 - 71*a - 40, 11*a^5 + 11*a^4 - 89*a^3 - 91*a^2 + 157*a + 153])
gp (2.8): E = ellinit([a^2 + a - 1, 2*a^5 - 3*a^4 - 8*a^3 + 10*a^2 + 6*a - 3, -a^5 + 2*a^4 + 4*a^3 - 7*a^2 - 3*a + 4, -6*a^5 + 2*a^4 + 42*a^3 + 5*a^2 - 71*a - 40, 11*a^5 + 11*a^4 - 89*a^3 - 91*a^2 + 157*a + 153],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((7,-2 a^{5} + 3 a^{4} + 10 a^{3} - 12 a^{2} - 11 a + 6)\) | = | \( \left(-2 a^{5} + 3 a^{4} + 10 a^{3} - 12 a^{2} - 11 a + 6\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 49 \) | = | \( 49 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((49,49 a^{5} - 49 a^{4} - 245 a^{3} + 196 a^{2} + 245 a - 98,28 a^{5} - 27 a^{4} - 141 a^{3} + 109 a^{2} + 142 a - 17,19 a^{5} - 19 a^{4} - 94 a^{3} + 75 a^{2} + 92 a - 19,36 a^{5} - 36 a^{4} - 180 a^{3} + 144 a^{2} + 181 a - 54,5 a^{5} - 5 a^{4} - 25 a^{3} + 21 a^{2} + 25 a + 36)\) | = | \( \left(-2 a^{5} + 3 a^{4} + 10 a^{3} - 12 a^{2} - 11 a + 6\right)^{2} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 2401 \) | = | \( 49^{2} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -\frac{44682954498275}{49} a^{5} + \frac{20731677864676}{49} a^{4} + \frac{214541279828608}{49} a^{3} - \frac{20858262562702}{49} a^{2} - \frac{124233907036096}{49} a + \frac{28389749219352}{49} \) | ||
| 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: | $\left(-6 a^{5} + \frac{35}{4} a^{4} + \frac{55}{2} a^{3} - \frac{127}{4} a^{2} - \frac{55}{2} a + \frac{51}{4} : \frac{5}{8} a^{5} - \frac{13}{8} a^{3} + \frac{3}{4} a^{2} + \frac{3}{2} a - \frac{1}{4} : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-2 a^{5} + 3 a^{4} + 10 a^{3} - 12 a^{2} - 11 a + 6\right) \) | \(49\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) 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.