Base field 6.6.1241125.1
Generator \(a\), with minimal polynomial \( x^{6} - 7 x^{4} - 2 x^{3} + 11 x^{2} + 7 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 7, 11, -2, -7, 0, 1]))
gp: K = nfinit(Polrev([1, 7, 11, -2, -7, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 11, -2, -7, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-11,-29,0,20,1,-3]),K([11,32,5,-20,-2,3]),K([-3,-11,-3,7,1,-1]),K([212,891,105,-565,-45,84]),K([667,3103,365,-1998,-153,298])])
gp: E = ellinit([Polrev([-11,-29,0,20,1,-3]),Polrev([11,32,5,-20,-2,3]),Polrev([-3,-11,-3,7,1,-1]),Polrev([212,891,105,-565,-45,84]),Polrev([667,3103,365,-1998,-153,298])], K);
magma: E := EllipticCurve([K![-11,-29,0,20,1,-3],K![11,32,5,-20,-2,3],K![-3,-11,-3,7,1,-1],K![212,891,105,-565,-45,84],K![667,3103,365,-1998,-153,298]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^5-a^4-6a^3+3a^2+8a+4)\) | = | \((-2a^5+a^4+13a^3-2a^2-19a-5)\cdot(2a^5-a^4-14a^3+2a^2+23a+6)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 45 \) | = | \(5\cdot9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-7a^5+2a^4+48a^3-a^2-71a-21)\) | = | \((-2a^5+a^4+13a^3-2a^2-19a-5)^{4}\cdot(2a^5-a^4-14a^3+2a^2+23a+6)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 50625 \) | = | \(5^{4}\cdot9^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{111388131328}{225} a^{5} + \frac{62867896343}{225} a^{4} + \frac{155339260534}{45} a^{3} - \frac{157390329971}{225} a^{2} - \frac{1225752719447}{225} a - \frac{50545458893}{45} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(0\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(\frac{9}{4} a^{5} - a^{4} - 15 a^{3} + 2 a^{2} + \frac{91}{4} a + \frac{5}{2} : -\frac{29}{8} a^{5} + \frac{7}{8} a^{4} + \frac{189}{8} a^{3} + \frac{11}{8} a^{2} - \frac{131}{4} a - 12 : 1\right)$ | $\left(2 a^{5} - 15 a^{3} + 23 a + 3 : -3 a^{5} + 20 a^{3} + 3 a^{2} - 27 a - 10 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 3849.5526445693109051125313105813306236 \) | ||
| Tamagawa product: | \( 8 \) = \(2^{2}\cdot2\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 1.72772 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^5+a^4+13a^3-2a^2-19a-5)\) | \(5\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((2a^5-a^4-14a^3+2a^2+23a+6)\) | \(9\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
45.1-b
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.