Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+4008849x+3451333210\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+4008849xz^2+3451333210z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+4008849x+3451333210\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-754, 0)$ | $0$ | $2$ |
Integral points
\( \left(-754, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 413712 \) | = | $2^{4} \cdot 3^{2} \cdot 13^{2} \cdot 17$ |
|
| Discriminant: | $\Delta$ | = | $-9269099110659922350336$ | = | $-1 \cdot 2^{8} \cdot 3^{12} \cdot 13^{8} \cdot 17^{4} $ |
|
| j-invariant: | $j$ | = | \( \frac{7909612346288}{10289870721} \) | = | $2^{4} \cdot 3^{-6} \cdot 13^{-2} \cdot 17^{-4} \cdot 7907^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9009801602188904083074754373$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.60710121678077032163828768375$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9326517624097936$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.441933809016545$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.087256926993167357278232420441$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2\cdot2^{2}\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $0.69805541594533885822585936353 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 0.698055416 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.087257 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 0.698055416\end{aligned}$$
Modular invariants
Modular form 413712.2.a.bk
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18579456 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_0^{*}$ | additive | -1 | 4 | 8 | 0 |
| $3$ | $2$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $13$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $17$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2B | 8.12.0.38 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1768 = 2^{3} \cdot 13 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3 & 8 \\ 10 & 27 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1761 & 8 \\ 1760 & 9 \end{array}\right),\left(\begin{array}{rr} 105 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1767 & 1760 \\ 1766 & 1751 \end{array}\right),\left(\begin{array}{rr} 885 & 8 \\ 2 & 17 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 1325 & 1764 \\ 663 & 1767 \end{array}\right),\left(\begin{array}{rr} 271 & 1762 \\ 0 & 1767 \end{array}\right)$.
The torsion field $K:=\Q(E[1768])$ is a degree-$65696956416$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1768\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | additive | $2$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
| $3$ | additive | $2$ | \( 45968 = 2^{4} \cdot 13^{2} \cdot 17 \) |
| $13$ | additive | $98$ | \( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 24336 = 2^{4} \cdot 3^{2} \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 413712.bk
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 2652.a2, its twist by $156$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.