Base field \(\Q(\sqrt{57}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 14 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{2}{9} a - \frac{7}{9} : \frac{5}{9} a - \frac{11}{27} : 1\right)$ | $1.5796393556378250833431632301533697124$ | $\infty$ |
| $\left(\frac{1}{4} a + \frac{1}{4} : -\frac{3}{8} a - \frac{15}{8} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((7a+19)\) | = | \((a+3)^{6}\cdot(4a+13)\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 192 \) | = | \(2^{6}\cdot3\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $3871a-5429$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((3871a-5429)\) | = | \((a+3)^{26}\cdot(4a+13)\) |
|
| |||||
| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( -201326592 \) | = | \(-2^{26}\cdot3\) |
|
| |||||
| j-invariant: | $j$ | = | \( \frac{152551}{768} a + \frac{476915}{384} \) | ||
|
| |||||
| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | ||
|
| |||||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 1 \) |
|
|
|||
| Mordell-Weil rank: | $r$ | = | \(1\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.5796393556378250833431632301533697124 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 3.1592787112756501666863264603067394248 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 7.3446892855594700344355639234373458686 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2^{2}\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.0734343514077692234767500123389828889 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.073434351 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 7.344689 \cdot 3.159279 \cdot 4 } { {2^2 \cdot 7.549834} } \\ & \approx 3.073434351 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a+3)\) | \(2\) | \(4\) | \(I_{16}^{*}\) | Additive | \(1\) | \(6\) | \(26\) | \(8\) |
| \((4a+13)\) | \(3\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
192.6-l
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.