Base field \(\Q(\sqrt{42}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 42 \); class number \(2\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.
Invariants
Conductor: | \((8,4a)\) | = | \((2,a)^{5}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 32 \) | = | \(2^{5}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-4096)\) | = | \((2,a)^{24}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 16777216 \) | = | \(2^{24}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((64)\) | = | \((2,a)^{12}\) |
Minimal discriminant norm: | \( 4096 \) | = | \(2^{12}\) |
j-invariant: | \( 1728 \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z[\sqrt{-1}]\) | (potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $N(\mathrm{U}(1))$ |
Mordell-Weil group
Rank: | \(2\) | |
Generators | $\left(-\frac{4}{7} a + \frac{26}{7} : \frac{540}{49} a - \frac{500}{7} : 1\right)$ | $\left(-\frac{16}{3} a + \frac{104}{3} : -\frac{500}{9} a + 360 : 1\right)$ |
Heights | \(1.4944407538016703049134574705638987676\) | \(1.7772517496792384795883746699613785731\) |
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);
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Torsion generator: | $\left(0 : 0 : 1\right)$ | |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 2 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(2\) | ||
Regulator: | \( 2.6559974444859786139759566740605797568 \) | ||
Period: | \( 13.750371636040745654980191559621114396 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 5.6353052259950623930642587390953015321 \) | ||
Analytic order of Ш: | \( 1 \) (rounded) |
Local data at primes of bad reduction
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(2\) | \(III^{*}\) | Additive | \(1\) | \(5\) | \(12\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
The image is a Borel subgroup if \(p=2\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
32.1-d
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 576.c4 |
\(\Q\) | 1568.e4 |