Base field \(\Q(\sqrt{-182}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 182 \); class number \(12\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((13,a)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-104 : -78 a : 1\right)$ | $2.1341208490447540200516508019456646351$ | $\infty$ |
| $\left(\frac{4225}{36} : \frac{272935}{216} : 1\right)$ | $5.8325117369876505579512553123738599886$ | $\infty$ |
| $\left(-13 : 0 : 1\right)$ | $0$ | $2$ |
| $\left(0 : 0 : 1\right)$ | $0$ | $2$ |
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: | $\Delta$ | = | $308915776$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((308915776)\) | = | \((2,a)^{12}\cdot(13,a)^{12}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 95428956661682176 \) | = | \(2^{12}\cdot13^{12}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((64)\) | = | \((2,a)^{12}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 4096 \) | = | \(2^{12}\) |
| j-invariant: | $j$ | = | \( 1728 \) | ||
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[\sqrt{-1}]\) (potential complex multiplication) | ||
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 2 \) |
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| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 12.447284900203577858478890649255537485 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 49.789139600814311433915562597022149940 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 27.500743272081491309960383119242228792 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 2 \) = \(2\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 12.686837353055228805085629542111911397 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 4 \) (rounded) |
BSD formula
$$\begin{aligned}12.686837353 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 4 \cdot 13.750372 \cdot 49.789140 \cdot 2 } { {4^2 \cdot 26.981475} } \\ & \approx 12.686837353 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(2\) | \(I_{3}^{*}\) | Additive | \(-1\) | \(5\) | \(12\) | \(0\) |
| \((13,a)\) | \(13\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
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.
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 |
|---|---|
| \(\Q\) | 3136.m3 |
| \(\Q\) | 5408.g3 |