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([1/6,-1/3,0,1/6]),K([-49/6,1/3,1,-1/6]),K([7/6,-1/3,0,1/6]),K([-723/2,-87,35,15/2]),K([7489/6,3434/3,-103,-671/6])])
gp: E = ellinit([Polrev([1/6,-1/3,0,1/6]),Polrev([-49/6,1/3,1,-1/6]),Polrev([7/6,-1/3,0,1/6]),Polrev([-723/2,-87,35,15/2]),Polrev([7489/6,3434/3,-103,-671/6])], K);
magma: E := EllipticCurve([K![1/6,-1/3,0,1/6],K![-49/6,1/3,1,-1/6],K![7/6,-1/3,0,1/6],K![-723/2,-87,35,15/2],K![7489/6,3434/3,-103,-671/6]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-1/3a^3-a^2+8/3a+20/3)\) | = | \((-a+3)\cdot(1/2a^3+a^2-5a-21/2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 20 \) | = | \(4\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((5a+5)\) | = | \((-a+3)\cdot(1/2a^3+a^2-5a-21/2)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 12500 \) | = | \(4\cdot5^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{385810183859}{50} a^{3} - \frac{565777507637}{50} a^{2} + \frac{4001928738953}{50} a + \frac{3202354723143}{25} \) | ||
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 | $\left(\frac{1}{6} a^{3} - 3 a^{2} - \frac{4}{3} a + \frac{175}{6} : \frac{17}{6} a^{3} - 5 a^{2} - \frac{86}{3} a + \frac{287}{6} : 1\right)$ |
| Height | \(0.26017150632276335782970590683547784978\) |
| 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: | $\left(-\frac{17}{24} a^{3} - a^{2} + \frac{89}{12} a + \frac{205}{24} : \frac{19}{24} a^{3} + \frac{7}{8} a^{2} - \frac{167}{24} a - \frac{65}{6} : 1\right)$ |
| 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: | \( 0.26017150632276335782970590683547784978 \) | ||
| Period: | \( 380.17925964657120673240314181812312393 \) | ||
| Tamagawa product: | \( 5 \) = \(1\cdot5\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.62384835435329 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+3)\) | \(4\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((1/2a^3+a^2-5a-21/2)\) | \(5\) | \(5\) | \(I_{5}\) | 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 |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
20.1-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.