Base field \(\Q(\sqrt{-3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-4 : 45 a - 23 : 1\right)$ | $0.12003476494896236576467112446821754487$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-290a+145)\) | = | \((-2a+1)\cdot(5)\cdot(29)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 63075 \) | = | \(3\cdot25\cdot841\) |
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| Discriminant: | $\Delta$ | = | $-550546875$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-550546875)\) | = | \((-2a+1)^{10}\cdot(5)^{7}\cdot(29)\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 303101861572265625 \) | = | \(3^{10}\cdot25^{7}\cdot841\) |
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| j-invariant: | $j$ | = | \( \frac{53838872576}{550546875} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.12003476494896236576467112446821754487 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.240069529897924731529342248936435089740 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 0.976829287538608030773935600549861193440 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 14 \) = \(2\cdot7\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 1.8954970925518306660577331284133436021 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}1.895497093 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.976829 \cdot 0.240070 \cdot 14 } { {1^2 \cdot 1.732051} } \\ & \approx 1.895497093 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 3 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(3\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
| \((5)\) | \(25\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
| \((29)\) | \(841\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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 63075.1-a consists of this curve only.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 435.b1 |
| \(\Q\) | 1305.c1 |