Base field \(\Q(\sqrt{-69}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 69 \); class number \(8\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((13,a+3)\). 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 |
|---|---|---|
| $\left(\frac{219}{169} a - \frac{1514}{169} : \frac{1874}{2197} a + \frac{133529}{2197} : 1\right)$ | $4.3414109465521889926368271283146777107$ | $\infty$ |
| $\left(\frac{3}{2} a - 11 : \frac{17}{4} a + \frac{227}{4} : 1\right)$ | $0$ | $2$ |
| $\left(a - 6 : 2 a + 37 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((12,2a+6)\) | = | \((2,a+1)^{3}\cdot(3,a)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 24 \) | = | \(2^{3}\cdot3\) |
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| Discriminant: | $\Delta$ | = | $640664640a-3287963664$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((640664640a-3287963664)\) | = | \((2,a+1)^{8}\cdot(3,a)^{8}\cdot(13,a+3)^{12}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 39131836541081047296 \) | = | \(2^{8}\cdot3^{8}\cdot13^{12}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((1296)\) | = | \((2,a+1)^{8}\cdot(3,a)^{8}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 1679616 \) | = | \(2^{8}\cdot3^{8}\) |
| j-invariant: | $j$ | = | \( \frac{1556068}{81} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 4.3414109465521889926368271283146777107 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 8.6828218931043779852736542566293554214 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 7.2706940358628777611249866685015359952 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 32 \) = \(2^{2}\cdot2^{3}\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 7.5999759221644019338766070427064258958 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}7.599975922 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 7.270694 \cdot 8.682822 \cdot 32 } { {4^2 \cdot 16.613248} } \\ & \approx 7.599975922 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a+1)\) | \(2\) | \(4\) | \(I_{1}^{*}\) | Additive | \(-1\) | \(3\) | \(8\) | \(0\) |
| \((3,a)\) | \(3\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
| \((13,a+3)\) | \(13\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
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
24.1-b
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 |
|---|---|
| \(\Q\) | 72.a3 |
| \(\Q\) | 25392.be3 |