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.
Mordell-Weil group structure
Not computed ($ 0 \le r \le 2 $)
Mordell-Weil generators
No non-torsion generators are known.
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(1 : -2 : 1\right)$ | $0$ | $4$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((8,4a)\) | = | \((2,a)^{5}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 32 \) | = | \(2^{5}\) |
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| Discriminant: | $\Delta$ | = | $512$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((512)\) | = | \((2,a)^{18}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 262144 \) | = | \(2^{18}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((8)\) | = | \((2,a)^{6}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 64 \) | = | \(2^{6}\) |
| j-invariant: | $j$ | = | \( 287496 \) | ||
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[\sqrt{-4}]\) (potential complex multiplication) | ||
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 0 \) |
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| Mordell-Weil rank: | $r?$ | \(0 \le r \le 2\) | |
| Regulator*: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1 \) |
| Néron-Tate Regulator*: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 1 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 55.001486544162982619920766238484457583 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 2 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.1217284066648174691390538064525657647 \) |
| Analytic order of Ш*: | Ш${}_{\mathrm{an}}$ | = | \( 4 \) (rounded) |
* Conditional on BSD: assuming rank = analytic rank.
Note: We expect that the nontriviality of Ш explains the discrepancy between the upper bound on the rank and the analytic rank. The application of further descents should suffice to establish the weak BSD conjecture for this curve.
Local data at primes of bad reduction
This elliptic curve is not semistable. There is only one prime $\frak{p}$ of bad reduction. Primes of good reduction for the curve but which divide the discriminant of the model above (if any) are included.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(2\) | \(III\) | Additive | \(1\) | \(5\) | \(6\) | \(0\) |
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-a
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 |
|---|---|
| \(\Q\) | 32.a2 |
| \(\Q\) | 28224.fc2 |