Base field 4.4.13888.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 7 x^{2} + 6 x + 9 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([9, 6, -7, -2, 1]))
gp: K = nfinit(Polrev([9, 6, -7, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 6, -7, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-4,0,1,0]),K([4,7/3,-1/3,-1/3]),K([2,-1/3,-2/3,1/3]),K([-26,49/3,26/3,-10/3]),K([39,-16,-11,4])])
gp: E = ellinit([Polrev([-4,0,1,0]),Polrev([4,7/3,-1/3,-1/3]),Polrev([2,-1/3,-2/3,1/3]),Polrev([-26,49/3,26/3,-10/3]),Polrev([39,-16,-11,4])], K);
magma: E := EllipticCurve([K![-4,0,1,0],K![4,7/3,-1/3,-1/3],K![2,-1/3,-2/3,1/3],K![-26,49/3,26/3,-10/3],K![39,-16,-11,4]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^2+a+5)\) | = | \((1/3a^3-2/3a^2-4/3a+1)\cdot(1/3a^3-2/3a^2-7/3a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 28 \) | = | \(4\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((192a^3+769a^2-2653a-6679)\) | = | \((1/3a^3-2/3a^2-4/3a+1)\cdot(1/3a^3-2/3a^2-7/3a+1)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -67228 \) | = | \(-4\cdot7^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{5814380907}{33614} a^{3} + \frac{16966182151}{33614} a^{2} + \frac{25433678299}{33614} a - \frac{29096936969}{16807} \) | ||
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 | $\left(-\frac{5}{6} a^{3} + \frac{13}{6} a^{2} + \frac{10}{3} a - \frac{15}{2} : -\frac{1}{4} a^{3} + 2 a^{2} + \frac{1}{2} a - \frac{27}{4} : 1\right)$ |
| Height | \(0.79018609310187034254150770329701755598\) |
| Torsion structure: | \(\Z/5\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(0 : \frac{2}{3} a^{3} - \frac{4}{3} a^{2} - \frac{11}{3} a + 4 : 1\right)$ |
| 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: | \( 0.79018609310187034254150770329701755598 \) | ||
| Period: | \( 602.07121776438061863079375635130230058 \) | ||
| Tamagawa product: | \( 5 \) = \(1\cdot5\) | ||
| Torsion order: | \(5\) | ||
| Leading coefficient: | \( 3.22958992221777 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/3a^3-2/3a^2-4/3a+1)\) | \(4\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((1/3a^3-2/3a^2-7/3a+1)\) | \(7\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(5\) | 5B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
28.1-a
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.