Base field 4.4.16317.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 5, -4, -2, 1]))
gp: K = nfinit(Polrev([1, 5, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0,0]),K([-5,4,2,-1]),K([2,-4,-1,1]),K([18,-11,-7,3]),K([-68,43,26,-12])])
gp: E = ellinit([Polrev([1,1,0,0]),Polrev([-5,4,2,-1]),Polrev([2,-4,-1,1]),Polrev([18,-11,-7,3]),Polrev([-68,43,26,-12])], K);
magma: E := EllipticCurve([K![1,1,0,0],K![-5,4,2,-1],K![2,-4,-1,1],K![18,-11,-7,3],K![-68,43,26,-12]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^3+a^2+5a-2)\) | = | \((a+1)\cdot(-a^3+2a^2+3a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 35 \) | = | \(5\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-3a^3+4a^2+12a+5)\) | = | \((a+1)^{4}\cdot(-a^3+2a^2+3a-3)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 4375 \) | = | \(5^{4}\cdot7\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1323075862}{4375} a^{3} - \frac{1061976641}{4375} a^{2} - \frac{6527302567}{4375} a - \frac{1115839118}{4375} \) | ||
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: | \(2\) | |
| Generators | $\left(\frac{7}{3} a^{3} - \frac{16}{3} a^{2} - \frac{25}{3} a + \frac{41}{3} : -\frac{82}{9} a^{3} + \frac{59}{3} a^{2} + \frac{299}{9} a - \frac{458}{9} : 1\right)$ | $\left(a^{3} - 2 a^{2} - 4 a + 4 : -a^{3} + 2 a^{2} + 4 a - 5 : 1\right)$ |
| Heights | \(0.66909682246886232424290586682920941718\) | \(0.31686156707864516186584743729838131270\) |
| Torsion structure: | trivial | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
BSD invariants
| Analytic rank: | \( 2 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
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| Mordell-Weil rank: | \(2\) | ||
| Regulator: | \( 0.017720437168688465500687502047147493916 \) | ||
| Period: | \( 1208.2523383639373322575618761620061852 \) | ||
| Tamagawa product: | \( 2 \) = \(2\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 5.36366811919881 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+1)\) | \(5\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
| \((-a^3+2a^2+3a-3)\) | \(7\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 35.2-b consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.