Base field \(\Q(\sqrt{-42}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 42 \); class number \(4\).
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
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{641}{56} : -\frac{9573}{1568} a : 1\right)$ | $5.2770176462683163497710067801195825324$ | $\infty$ |
| $\left(-13 : 6 a : 1\right)$ | $0$ | $2$ |
| $\left(11 : -6 a : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((21,a)\) | = | \((3,a)\cdot(7,a)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 21 \) | = | \(3\cdot7\) |
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| Discriminant: | $\Delta$ | = | $1382976$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((1382976)\) | = | \((2,a)^{12}\cdot(3,a)^{4}\cdot(7,a)^{8}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 1912622616576 \) | = | \(2^{12}\cdot3^{4}\cdot7^{8}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((21609)\) | = | \((3,a)^{4}\cdot(7,a)^{8}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 466948881 \) | = | \(3^{4}\cdot7^{8}\) |
| j-invariant: | $j$ | = | \( \frac{13027640977}{21609} \) | ||
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | ||
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 1 \) |
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| Mordell-Weil rank: | $r$ | = | \(1\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 5.2770176462683163497710067801195825324 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 10.554035292536632699542013560239165065 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 3.4483077185485079225446290683660727928 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 32 \) = \(1\cdot2^{2}\cdot2^{3}\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 5.6156484350667397494729808867983361621 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}5.615648435 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 3.448308 \cdot 10.554035 \cdot 32 } { {4^2 \cdot 12.961481} } \\ & \approx 5.615648435 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 2 primes $\frak{p}$ 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((3,a)\) | \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((7,a)\) | \(7\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
21.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 441.f2 |
| \(\Q\) | 1344.s2 |