Base field \(\Q(\sqrt{-2}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 2 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{8176}{361} a - \frac{16127}{722} : -\frac{2080851}{27436} a + \frac{1227905}{13718} : 1\right)$ | $6.2931539160546365928133690826298624376$ | $\infty$ |
| $\left(-16 a - \frac{73}{2} : \frac{73}{4} a - 16 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((204)\) | = | \((a)^{4}\cdot(-a-1)\cdot(a-1)\cdot(-2a+3)\cdot(2a+3)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 41616 \) | = | \(2^{4}\cdot3\cdot3\cdot17\cdot17\) |
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| Discriminant: | $\Delta$ | = | $1788724321920a+1783310266368$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((1788724321920a+1783310266368)\) | = | \((a)^{15}\cdot(-a-1)^{6}\cdot(a-1)^{24}\cdot(-2a+3)^{4}\cdot(2a+3)\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 9579264905789834696884224 \) | = | \(2^{15}\cdot3^{6}\cdot3^{24}\cdot17^{4}\cdot17\) |
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| j-invariant: | $j$ | = | \( -\frac{410311536960329978125}{94355189265718404} a - \frac{136155893118005119000}{23588797316429601} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 6.2931539160546365928133690826298624376 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 12.586307832109273185626738165259724875 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 0.207933967460669588201616490057428731420 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 16 \) = \(2\cdot2\cdot2\cdot2\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.7011679039764952941272628899277629300 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.701167904 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.207934 \cdot 12.586308 \cdot 16 } { {2^2 \cdot 2.828427} } \\ & \approx 3.701167904 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(2\) | \(I_{7}^{*}\) | Additive | \(1\) | \(4\) | \(15\) | \(3\) |
| \((-a-1)\) | \(3\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
| \((a-1)\) | \(3\) | \(2\) | \(I_{24}\) | Non-split multiplicative | \(1\) | \(1\) | \(24\) | \(24\) |
| \((-2a+3)\) | \(17\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
| \((2a+3)\) | \(17\) | \(1\) | \(I_{1}\) | Non-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 |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
41616.5-e
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.