Base field 4.4.18625.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 14 x^{2} + 9 x + 41 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([41, 9, -14, -1, 1]))
gp: K = nfinit(Polrev([41, 9, -14, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 9, -14, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-6,0,1,0]),K([6,-1,-1,0]),K([-7,0,1,0]),K([-148/3,-418/3,2,41/3]),K([-4267/6,-1274/3,63,245/6])])
gp: E = ellinit([Polrev([-6,0,1,0]),Polrev([6,-1,-1,0]),Polrev([-7,0,1,0]),Polrev([-148/3,-418/3,2,41/3]),Polrev([-4267/6,-1274/3,63,245/6])], K);
magma: E := EllipticCurve([K![-6,0,1,0],K![6,-1,-1,0],K![-7,0,1,0],K![-148/3,-418/3,2,41/3],K![-4267/6,-1274/3,63,245/6]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/2a^3+a^2-5a-21/2)\) | = | \((1/2a^3+a^2-5a-21/2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 5 \) | = | \(5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((25)\) | = | \((1/2a^3+a^2-5a-21/2)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 390625 \) | = | \(5^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{10364321917}{150} a^{3} - \frac{503906304}{5} a^{2} + \frac{10801512773}{15} a + \frac{172670364179}{150} \) | ||
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 | $\left(\frac{2}{3} a^{3} + a^{2} - \frac{19}{3} a - \frac{22}{3} : -\frac{1}{6} a^{3} - 5 a^{2} - \frac{5}{3} a + \frac{227}{6} : 1\right)$ |
| Height | \(0.57768645107962011198383452552595973650\) |
| 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: | $\left(\frac{5}{12} a^{3} + \frac{1}{2} a^{2} - \frac{49}{12} a - \frac{73}{12} : -\frac{1}{12} a^{3} - \frac{1}{2} a^{2} + \frac{5}{12} a + \frac{97}{24} : 1\right)$ |
| 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: | \( 0.57768645107962011198383452552595973650 \) | ||
| Period: | \( 66.160913572455727379546333958272267014 \) | ||
| Tamagawa product: | \( 8 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.24045136886674 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/2a^3+a^2-5a-21/2)\) | \(5\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
5.1-b
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.