Base field \(\Q(\sqrt{-111}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 28 \); class number \(8\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((3,a+1)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{847}{3} : \frac{6424}{9} a - \frac{1946}{9} : 1\right)$ | $5.6341391709079743635462328206703939278$ | $\infty$ |
| $\left(75 : -290 : 1\right)$ | $0$ | $6$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((14)\) | = | \((2,a)\cdot(2,a+1)\cdot(7,a)\cdot(7,a+6)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 196 \) | = | \(2\cdot2\cdot7\cdot7\) |
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| Discriminant: | $\Delta$ | = | $18289152$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((18289152)\) | = | \((2,a)^{9}\cdot(2,a+1)^{9}\cdot(3,a+1)^{12}\cdot(7,a)^{2}\cdot(7,a+6)^{2}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 334493080879104 \) | = | \(2^{9}\cdot2^{9}\cdot3^{12}\cdot7^{2}\cdot7^{2}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((25088)\) | = | \((2,a)^{9}\cdot(2,a+1)^{9}\cdot(7,a)^{2}\cdot(7,a+6)^{2}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 629407744 \) | = | \(2^{9}\cdot2^{9}\cdot7^{2}\cdot7^{2}\) |
| j-invariant: | $j$ | = | \( \frac{2251439055699625}{25088} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 5.6341391709079743635462328206703939278 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 11.26827834181594872709246564134078786 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 1.750834270388010790998569979249046268 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 324 \) = \(3^{2}\cdot3^{2}\cdot1\cdot2\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(6\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 8.4266242592693044386737431176757949441 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}8.426624259 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.875417 \cdot 11.268278 \cdot 324 } { {6^2 \cdot 10.535654} } \\ & \approx 8.426624259 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 4 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
| \((2,a+1)\) | \(2\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
| \((3,a+1)\) | \(3\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((7,a)\) | \(7\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((7,a+6)\) | \(7\) | \(2\) | \(I_{2}\) | 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 |
|---|---|
| \(2\) | 2B |
| \(3\) | 3Cs.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 6, 9 and 18.
Its isogeny class
196.5-f
consists of curves linked by isogenies of
degrees dividing 18.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 126.b1 |
| \(\Q\) | 19166.a1 |