Base field \(\Q(\sqrt{42}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 42 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-42, 0, 1]))
gp: K = nfinit(Polrev([-42, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-42, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,1]),K([0,0]),K([-486,-72]),K([-9762,-1492])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,1]),Polrev([0,0]),Polrev([-486,-72]),Polrev([-9762,-1492])], K);
magma: E := EllipticCurve([K![0,1],K![1,1],K![0,0],K![-486,-72],K![-9762,-1492]]);
This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a-6)\) | = | \((2,a)\cdot(3,a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 6 \) | = | \(2\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((524288a-3145728)\) | = | \((2,a)^{39}\cdot(3,a)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( -1649267441664 \) | = | \(-2^{39}\cdot3\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
Minimal discriminant: | \((8192a-49152)\) | = | \((2,a)^{27}\cdot(3,a)\) |
Minimal discriminant norm: | \( -402653184 \) | = | \(-2^{27}\cdot3\) |
j-invariant: | \( -\frac{76452242473}{49152} a - \frac{82500763787}{8192} \) | ||
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 | $\left(\frac{2624253}{104329} a + \frac{16553867}{104329} : -\frac{18020900704}{33698267} a - \frac{117193892784}{33698267} : 1\right)$ |
Height | \(9.3515988922530547088418173754932047684\) |
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: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
|
|||
Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 9.3515988922530547088418173754932047684 \) | ||
Period: | \( 1.7516958885044089836458623512359106783 \) | ||
Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.5276674523525468864084949120403668762 \) | ||
Analytic order of Ш: | \( 1 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
Primes of good reduction for the curve but which divide the
discriminant of the model above (if any) are included.
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(1\) | \(I_{27}\) | Non-split multiplicative | \(1\) | \(1\) | \(27\) | \(27\) |
\((3,a)\) | \(3\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
6.1-c
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.