Base field 5.5.24217.1
Generator \(a\), with minimal polynomial \( x^{5} - 5 x^{3} - x^{2} + 3 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 3, -1, -5, 0, 1]))
gp: K = nfinit(Polrev([1, 3, -1, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -1, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,1,5,0,-1]),K([-5,-2,14,1,-3]),K([-1,1,5,0,-1]),K([-57,-20,104,10,-22]),K([178,46,-329,-24,68])])
gp: E = ellinit([Polrev([-1,1,5,0,-1]),Polrev([-5,-2,14,1,-3]),Polrev([-1,1,5,0,-1]),Polrev([-57,-20,104,10,-22]),Polrev([178,46,-329,-24,68])], K);
magma: E := EllipticCurve([K![-1,1,5,0,-1],K![-5,-2,14,1,-3],K![-1,1,5,0,-1],K![-57,-20,104,10,-22],K![178,46,-329,-24,68]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^4+2a^3+10a^2-5a-5)\) | = | \((-2a^4+2a^3+10a^2-5a-5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 73 \) | = | \(73\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-2a^4-a^3+8a^2+7a-3)\) | = | \((-2a^4+2a^3+10a^2-5a-5)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -5329 \) | = | \(-73^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{897914919792598}{5329} a^{4} - \frac{325870982359061}{5329} a^{3} - \frac{4373494381863622}{5329} a^{2} + \frac{691699015294266}{5329} a + \frac{2451613630132198}{5329} \) | ||
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 | $\left(4 a^{4} - a^{3} - 18 a^{2} + a + 8 : 2 a^{4} - 2 a^{3} - 13 a^{2} + 5 a + 9 : 1\right)$ |
| Height | \(0.027170887465636756351649465338283882937\) |
| 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: | $\left(\frac{13}{4} a^{4} - \frac{3}{4} a^{3} - \frac{61}{4} a^{2} + \frac{1}{2} a + \frac{13}{2} : \frac{1}{2} a^{4} - \frac{7}{8} a^{3} - \frac{25}{8} a^{2} + \frac{27}{8} a + \frac{7}{2} : 1\right)$ |
| 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: | \( 0.027170887465636756351649465338283882937 \) | ||
| Period: | \( 4078.7942015595808877565239731989897805 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.78039127 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^4+2a^3+10a^2-5a-5)\) | \(73\) | \(2\) | \(I_{2}\) | Non-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 |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
73.1-b
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.