Base field 3.3.1957.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 9 x + 10 \); class number \(2\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10, -9, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 9*x + 10)
gp (2.8): K = nfinit(a^3 - a^2 - 9*a + 10);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a^2 + a - 6, a^2 - 7, a^2 + a - 6, 51895*a^2 + 99196*a - 178244, 33758168*a^2 + 64528849*a - 115947554]),K);
sage: E = EllipticCurve(K, [a^2 + a - 6, a^2 - 7, a^2 + a - 6, 51895*a^2 + 99196*a - 178244, 33758168*a^2 + 64528849*a - 115947554])
gp (2.8): E = ellinit([a^2 + a - 6, a^2 - 7, a^2 + a - 6, 51895*a^2 + 99196*a - 178244, 33758168*a^2 + 64528849*a - 115947554],K)
This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((11,10 a + 4)\) | = | \( \left(11, a - 4\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 11 \) | = | \( 11 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \((\Delta)\) | = | \((495616,a + 256626,a^{2} + 54588)\) | = | \( \left(2, a\right)^{12} \cdot \left(11, a - 4\right)^{2} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\Delta)\) | = | \( 495616 \) | = | \( 2^{12} \cdot 11^{2} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(\mathfrak{D}\) | = | \((121,a^{2} + 2 a - 13)\) | = | \( \left(11, a - 4\right)^{2} \) |
| \(N(\mathfrak{D})\) | = | \( 121 \) | = | \( 11^{2} \) |
| \(j\) | = | \( -\frac{94367604927778}{121} a^{2} - \frac{12216950994706}{121} a + \frac{835509871243821}{121} \) | ||
| 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: | $\left(-\frac{177}{4} a^{2} - \frac{337}{4} a + \frac{305}{2} : \frac{945}{8} a^{2} + \frac{1813}{8} a - \frac{1613}{4} : 1\right)$ |
| 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()
Primes of good reduction for the curve but which divide the
discriminant of the model above (if any) are included.
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(2, a\right) \) | \(2\) | \(1\) | \(I_{0}\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \( \left(11, a - 4\right) \) | \(11\) | \(2\) | \(I_{2}\) | 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, 4 and 8.
Its isogeny class
11.1-a
consists of curves linked by isogenies of
degrees dividing 8.