Base field 5.5.176684.1
Generator \(a\), with minimal polynomial \( x^{5} - 6 x^{3} - x^{2} + 5 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 5, -1, -6, 0, 1]))
gp: K = nfinit(Polrev([-1, 5, -1, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 5, -1, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([6,-7,-11,1,2]),K([-2,2,5,0,-1]),K([4,-5,-6,1,1]),K([6,-40,-39,6,7]),K([132,-485,-531,71,97])])
gp: E = ellinit([Polrev([6,-7,-11,1,2]),Polrev([-2,2,5,0,-1]),Polrev([4,-5,-6,1,1]),Polrev([6,-40,-39,6,7]),Polrev([132,-485,-531,71,97])], K);
magma: E := EllipticCurve([K![6,-7,-11,1,2],K![-2,2,5,0,-1],K![4,-5,-6,1,1],K![6,-40,-39,6,7],K![132,-485,-531,71,97]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^4-a^3+5a^2+4a-3)\) | = | \((-a^3+a^2+4a-2)\cdot(-a^4+5a^2+a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 16 \) | = | \(2\cdot8\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2a^4-2a^3-6a^2+4a-6)\) | = | \((-a^3+a^2+4a-2)^{3}\cdot(-a^4+5a^2+a-2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -32768 \) | = | \(-2^{3}\cdot8^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{4083203}{16} a^{4} + \frac{519035989}{16} a^{3} + \frac{584541887}{16} a^{2} - \frac{935812163}{16} a + \frac{86216709}{8} \) | ||
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: | \(1\) |
| Generator | $\left(-98 a^{4} - 72 a^{3} + 537 a^{2} + 490 a - 135 : 1904 a^{4} + 1385 a^{3} - 10414 a^{2} - 9483 a + 2614 : 1\right)$ |
| Height | \(0.58268274163500325318366009131231683341\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-2 a^{4} - \frac{7}{4} a^{3} + 12 a^{2} + \frac{41}{4} a - \frac{21}{4} : -\frac{5}{2} a^{4} - \frac{19}{8} a^{3} + 13 a^{2} + \frac{121}{8} a - \frac{5}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.58268274163500325318366009131231683341 \) | ||
| Period: | \( 686.00635928268303456681940897842989306 \) | ||
| Tamagawa product: | \( 2 \) = \(1\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.37739691 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^3+a^2+4a-2)\) | \(2\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
| \((-a^4+5a^2+a-2)\) | \(8\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
16.1-e
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.