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 \((5,a+1)\). No global minimal model exists.
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $\frak{N}$ | = | \((48,6a+24)\) | = | \((2,a)^{4}\cdot(2,a+1)\cdot(3,a+1)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 288 \) | = | \(2^{4}\cdot2\cdot3^{2}\) |
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| Discriminant: | $\Delta$ | = | $15520896a+635060736$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((15520896a+635060736)\) | = | \((2,a)^{11}\cdot(2,a+1)^{7}\cdot(3,a+1)^{8}\cdot(5,a+1)^{12}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 419904000000000000 \) | = | \(2^{11}\cdot2^{7}\cdot3^{8}\cdot5^{12}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((165888,10368a+41472)\) | = | \((2,a)^{11}\cdot(2,a+1)^{7}\cdot(3,a+1)^{8}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 1719926784 \) | = | \(2^{11}\cdot2^{7}\cdot3^{8}\) |
| j-invariant: | $j$ | = | \( \frac{293905}{128} a + \frac{3102191}{96} \) | ||
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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: | $r_{\mathrm{an}}$ | = | \( 0 \) |
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| Mordell-Weil rank: | $r$ | = | \(0\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | = | \( 1 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | = | \( 1 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 7.8161073103618982208245007791731508724 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 8 \) = \(2^{2}\cdot1\cdot2\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.9674882997158219753114083470929370068 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.967488300 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 3.908054 \cdot 1 \cdot 8 } { {1^2 \cdot 10.535654} } \\ & \approx 2.967488300 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(4\) | \(I_{3}^{*}\) | Additive | \(1\) | \(4\) | \(11\) | \(0\) |
| \((2,a+1)\) | \(2\) | \(1\) | \(I_{7}\) | Non-split multiplicative | \(1\) | \(1\) | \(7\) | \(7\) |
| \((3,a+1)\) | \(3\) | \(2\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
| \((5,a+1)\) | \(5\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
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 288.2-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.