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 \((7,a+6)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(3 a + 11 : -83 a - 246 : 1\right)$ | $0.26161701541220849531095010975998485621$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((6,2a+2)\) | = | \((2,a)\cdot(2,a+1)\cdot(3,a+1)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 12 \) | = | \(2\cdot2\cdot3\) |
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| Discriminant: | $\Delta$ | = | $79725330432a+11795624361984$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((79725330432a+11795624361984)\) | = | \((2,a)^{25}\cdot(2,a+1)^{25}\cdot(3,a+1)^{2}\cdot(7,a+6)^{12}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 140255135731691135032098816 \) | = | \(2^{25}\cdot2^{25}\cdot3^{2}\cdot7^{12}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((100663296)\) | = | \((2,a)^{25}\cdot(2,a+1)^{25}\cdot(3,a+1)^{2}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 10133099161583616 \) | = | \(2^{25}\cdot2^{25}\cdot3^{2}\) |
| j-invariant: | $j$ | = | \( -\frac{136511322949}{100663296} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.26161701541220849531095010975998485621 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.523234030824416990621900219519969712420 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 1.24997406255616407482588106021614542076 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 50 \) = \(1\cdot5^{2}\cdot2\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.1038841182501003752092124247305885500 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.103884118 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1.249974 \cdot 0.523234 \cdot 50 } { {1^2 \cdot 10.535654} } \\ & \approx 3.103884118 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(1\) | \(I_{25}\) | Non-split multiplicative | \(1\) | \(1\) | \(25\) | \(25\) |
| \((2,a+1)\) | \(2\) | \(25\) | \(I_{25}\) | Split multiplicative | \(-1\) | \(1\) | \(25\) | \(25\) |
| \((3,a+1)\) | \(3\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((7,a+6)\) | \(7\) | \(1\) | \(I_0\) | 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 |
|---|---|
| \(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
12.2-b
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.