Base field \(\Q(\sqrt{-111}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 28 \); class number \(8\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{20840921}{4831204} a + \frac{34632809}{2070516} : -\frac{46779965591}{95570877528} a - \frac{7892122139}{13652982504} : 1\right)$ | $8.5765039357667483544080303611392283157$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-5a-7)\) | = | \((2,a+1)^{4}\cdot(7,a)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 784 \) | = | \(2^{4}\cdot7^{2}\) |
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| Discriminant: | $\Delta$ | = | $2115a+17857$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((2115a+17857)\) | = | \((2,a+1)^{12}\cdot(7,a)^{6}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 481890304 \) | = | \(2^{12}\cdot7^{6}\) |
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| j-invariant: | $j$ | = | \( 38477541376 \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 8.5765039357667483544080303611392283157 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 17.153007871533496708816060722278456631 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 1.99239151231836138690272937265243863084 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) = \(1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 1.6218978003486342228870811498454488835 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}1.621897800 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.996196 \cdot 17.153008 \cdot 1 } { {1^2 \cdot 10.535654} } \\ & \approx 1.621897800 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a+1)\) | \(2\) | \(1\) | \(II^{*}\) | Additive | \(-1\) | \(4\) | \(12\) | \(0\) |
| \((7,a)\) | \(7\) | \(1\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3Cn |
| \(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
784.13-a
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.