Base field \(\Q(\sqrt{-67}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 17 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{201}{4} : -\frac{71}{8} : 1\right)$ | $0.68748992539390365531758453633214711171$ | $\infty$ |
| $\left(\frac{7032823963}{2280226841764} a + \frac{1661341200948}{33532747673} : -\frac{64164181050497023}{1721619150295497044} a - \frac{49186666657966733}{202543429446529064} : 1\right)$ | $13.062308582484169451034106190310795122$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((67)\) | = | \((-2a+1)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 4489 \) | = | \(67^{2}\) |
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| Discriminant: | $\Delta$ | = | $300763$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((300763)\) | = | \((-2a+1)^{6}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 90458382169 \) | = | \(67^{6}\) |
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| j-invariant: | $j$ | = | \( -147197952000 \) | ||
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z[(1+\sqrt{-67})/2]\) (complex multiplication) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-67})/2]\) | ||
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{U}(1)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 2 \) |
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| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 7.9167601584284298354745849781642313058 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 31.667040633713719341898339912656925223 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 0.543303840643635077530006817398723873740 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 8.4076155433330238568021027136125937050 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}8.407615543 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.543304 \cdot 31.667041 \cdot 4 } { {1^2 \cdot 8.185353} } \\ & \approx 8.407615543 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There is only one prime $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(67\) | \(4\) | \(I_0^{*}\) | 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 |
|---|---|
| \(67\) | 67B.1.65[2] |
For all other primes \(p\), the image is a split Cartan subgroup if \(\left(\frac{ -67 }{p}\right)=+1\) or a nonsplit Cartan subgroup if \(\left(\frac{ -67 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has no rational isogenies other than endomorphisms. Its isogeny class 4489.1-CMa 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 |
|---|---|
| \(\Q\) | 4489.b1 |
| \(\Q\) | 4489.b2 |