Base field 4.4.19525.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 14 x^{2} + 15 x + 45 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([45, 15, -14, -2, 1]))
gp: K = nfinit(Polrev([45, 15, -14, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45, 15, -14, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0,0]),K([2,-8/3,-2/3,1/3]),K([1,1,0,0]),K([-11,-32/3,0,2/3]),K([62,71/3,-32/3,-1])])
gp: E = ellinit([Polrev([1,1,0,0]),Polrev([2,-8/3,-2/3,1/3]),Polrev([1,1,0,0]),Polrev([-11,-32/3,0,2/3]),Polrev([62,71/3,-32/3,-1])], K);
magma: E := EllipticCurve([K![1,1,0,0],K![2,-8/3,-2/3,1/3],K![1,1,0,0],K![-11,-32/3,0,2/3],K![62,71/3,-32/3,-1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((1/3a^3-a^2-7/3a+6)\) | = | \((1/3a^3-a^2-7/3a+6)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 19 \) | = | \(19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((-7/3a^3-14/3a^2+27a+53)\) | = | \((1/3a^3-a^2-7/3a+6)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( -130321 \) | = | \(-19^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( \frac{18434434191632}{390963} a^{3} + \frac{1698354377333}{20577} a^{2} - \frac{128992198435360}{390963} a - \frac{68454955252245}{130321} \) | ||
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 | $\left(-\frac{5}{9} a^{3} + \frac{38}{27} a^{2} + \frac{58}{27} a - \frac{20}{9} : -\frac{97}{81} a^{3} + \frac{227}{27} a^{2} - \frac{299}{81} a - \frac{898}{27} : 1\right)$ |
Height | \(2.0026521067832748016569998843974328194\) |
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: | $\left(-\frac{1}{3} a^{3} + \frac{5}{12} a^{2} + \frac{13}{6} a - \frac{1}{4} : \frac{7}{24} a^{3} + \frac{25}{24} a^{2} - \frac{95}{24} a - \frac{63}{8} : 1\right)$ |
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: | \( 2.0026521067832748016569998843974328194 \) | ||
Period: | \( 87.674049834438376075854966621208705723 \) | ||
Tamagawa product: | \( 4 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 5.02621485991580 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((1/3a^3-a^2-7/3a+6)\) | \(19\) | \(4\) | \(I_{4}\) | 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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
19.2-a
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.