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 \((5,a+3)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{1}{4} a - 6 : -\frac{19}{8} a + \frac{11}{2} : 1\right)$ | $0.15578154695944832165527438844879609962$ | $\infty$ |
| $\left(-\frac{7}{12} a - \frac{77}{12} : -\frac{239}{72} a + \frac{1627}{72} : 1\right)$ | $1.1168274737243714183469283141609281881$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((23,a+12)\) | = | \((23,a+12)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 23 \) | = | \(23\) |
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| Discriminant: | $\Delta$ | = | $401624a-8191083$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((401624a-8191083)\) | = | \((5,a+3)^{12}\cdot(23,a+12)^{4}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 68320556640625 \) | = | \(5^{12}\cdot23^{4}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((279841,a+21080)\) | = | \((23,a+12)^{4}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 279841 \) | = | \(23^{4}\) |
| j-invariant: | $j$ | = | \( -\frac{102662144}{279841} a + \frac{140804096}{279841} \) | ||
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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: | $r_{\mathrm{an}}$ | = | \( 2 \) |
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| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.14394684371286970845281335821131772758 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.575787374851478833811253432845270910320 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 9.4553616859757020452719862909705572522 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(1\cdot2^{2}\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.0669919536659242958521902289807046496 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.066991954 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 9.455362 \cdot 0.575787 \cdot 4 } { {1^2 \cdot 10.535654} } \\ & \approx 2.066991954 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((5,a+3)\) | \(5\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((23,a+12)\) | \(23\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 23.2-b consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.