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 \((7,a)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{18643}{100} : -\frac{72149}{1000} a + \frac{18643}{200} : 1\right)$ | $7.6262030820714687523617708814972662892$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((26,a)\) | = | \((2,a)\cdot(13,a)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 26 \) | = | \(2\cdot13\) |
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| Discriminant: | $\Delta$ | = | $783071744$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((783071744)\) | = | \((2,a)^{18}\cdot(7,a)^{12}\cdot(13,a)^{2}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 613201356251201536 \) | = | \(2^{18}\cdot7^{12}\cdot13^{2}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((6656)\) | = | \((2,a)^{18}\cdot(13,a)^{2}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 44302336 \) | = | \(2^{18}\cdot13^{2}\) |
| j-invariant: | $j$ | = | \( -\frac{10730978619193}{6656} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 7.6262030820714687523617708814972662892 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 15.252406164142937504723541762994532578 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 3.5877365236621924677672703379465154330 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2\cdot1\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.0562359479859346976637339446244797734 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}4.056235948 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1.793868 \cdot 15.252406 \cdot 4 } { {1^2 \cdot 26.981475} } \\ & \approx 4.056235948 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(2\) | \(I_{18}\) | Non-split multiplicative | \(1\) | \(1\) | \(18\) | \(18\) |
| \((7,a)\) | \(7\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((13,a)\) | \(13\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
26.1-b
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 1274.d1 |
| \(\Q\) | 10816.k1 |