Base field \(\Q(\sqrt{-71}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 18 \); class number \(7\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((3,a)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(2 a - 5 : a + 11 : 1\right)$ | $0.12849700380390013328208878683543449802$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((18,2a+16)\) | = | \((2,a)\cdot(2,a+1)\cdot(3,a+2)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 36 \) | = | \(2\cdot2\cdot3^{2}\) |
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| Discriminant: | $\Delta$ | = | $-26927802a+128044476$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((-26927802a+128044476)\) | = | \((2,a)^{25}\cdot(2,a+1)\cdot(3,a)^{12}\cdot(3,a+2)^{6}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 25999348907114496 \) | = | \(2^{25}\cdot2\cdot3^{12}\cdot3^{6}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((24461180928,2a+9695019796)\) | = | \((2,a)^{25}\cdot(2,a+1)\cdot(3,a+2)^{6}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 48922361856 \) | = | \(2^{25}\cdot2\cdot3^{6}\) |
| j-invariant: | $j$ | = | \( \frac{10456234965}{33554432} a + \frac{34842889817}{33554432} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.12849700380390013328208878683543449802 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.2569940076078002665641775736708689960 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 3.489969105140492371473010870601449595 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 25 \) = \(5^{2}\cdot1\cdot1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.6610645754619780739583862967946435316 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.661064575 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 3.489969 \cdot 0.256994 \cdot 25 } { {1^2 \cdot 8.426150} } \\ & \approx 2.661064575 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(25\) | \(I_{25}\) | Split multiplicative | \(-1\) | \(1\) | \(25\) | \(25\) |
| \((2,a+1)\) | \(2\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((3,a)\) | \(3\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((3,a+2)\) | \(3\) | \(1\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5 and 25.
Its isogeny class
36.6-b
consists of curves linked by isogenies of
degrees dividing 25.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.