Base field \(\Q(\sqrt{-39}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 10 \); class number \(4\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((3,a+1)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(0 : -3 : 1\right)$ | $1.1303356261795510066487525098806379240$ | $\infty$ |
| $\left(-\frac{9}{16} a + \frac{11}{16} : \frac{9}{64} a + \frac{117}{64} : 1\right)$ | $1.5499276249962272627153389192326210034$ | $\infty$ |
| $\left(\frac{3}{4} a + \frac{1}{2} : -\frac{3}{8} a - \frac{3}{4} : 1\right)$ | $0$ | $2$ |
| $\left(-4 : 3 a : 1\right)$ | $0$ | $4$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-2a+1)\) | = | \((3,a+1)\cdot(13,a+6)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 39 \) | = | \(3\cdot13\) |
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| Discriminant: | $\Delta$ | = | $28431$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((28431)\) | = | \((3,a+1)^{14}\cdot(13,a+6)^{2}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 808321761 \) | = | \(3^{14}\cdot13^{2}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((39)\) | = | \((3,a+1)^{2}\cdot(13,a+6)^{2}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 1521 \) | = | \(3^{2}\cdot13^{2}\) |
| j-invariant: | $j$ | = | \( \frac{12167}{39} \) | ||
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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: | $r_{\mathrm{an}}$ | = | \( 2 \) |
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| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.4325237555804154371827083089631259758 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 5.7300950223216617487308332358525039032 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 15.122360343414913938848635514050078231 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(8\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 0.86721967076469983461665422359621759656 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}0.867219671 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 15.122360 \cdot 5.730095 \cdot 4 } { {8^2 \cdot 6.244998} } \\ & \approx 0.867219671 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 2 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((3,a+1)\) | \(3\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((13,a+6)\) | \(13\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
39.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 117.a4 |
| \(\Q\) | 507.a4 |