Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-165199727x+274987174154\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-165199727xz^2+274987174154z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-165199727x+274987174154\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1694, 0)$ | $0$ | $2$ |
Integral points
\( \left(1694, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 474320 \) | = | $2^{4} \cdot 5 \cdot 7^{2} \cdot 11^{2}$ |
|
| Discriminant: | $\Delta$ | = | $255874324577155894665440000$ | = | $2^{8} \cdot 5^{4} \cdot 7^{14} \cdot 11^{9} $ |
|
| j-invariant: | $j$ | = | \( \frac{7020843884784}{3603000625} \) | = | $2^{4} \cdot 3^{3} \cdot 5^{-4} \cdot 7^{-8} \cdot 17^{3} \cdot 149^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.7594422111715078726713127592$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.52596756167177643912735728970$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0046868275068956$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.232096180583225$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.048784764505145695693670942030$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2^{2}\cdot2^{2}\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $3.5125030443704900899443078261 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
|
BSD formula
$$\begin{aligned} 3.512503044 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.048785 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 3.512503044\end{aligned}$$
Modular invariants
Modular form 474320.2.a.fm
For more coefficients, see the Downloads section to the right.
| Modular degree: | 97320960 |
|
| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
|
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 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $7$ | $4$ | $I_{8}^{*}$ | additive | -1 | 2 | 14 | 8 |
| $11$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
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 | 8.24.0.136 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \), index $192$, genus $9$, and generators
$\left(\begin{array}{rr} 4941 & 16 \\ 2728 & 325 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 6036 & 6105 \end{array}\right),\left(\begin{array}{rr} 7 & 4 \\ 68 & 39 \end{array}\right),\left(\begin{array}{rr} 3376 & 5 \\ 2291 & 16 \end{array}\right),\left(\begin{array}{rr} 5281 & 16 \\ 5288 & 129 \end{array}\right),\left(\begin{array}{rr} 8 & 1541 \\ 5389 & 4620 \end{array}\right),\left(\begin{array}{rr} 9 & 16 \\ 320 & 569 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 6145 & 16 \\ 6144 & 17 \end{array}\right),\left(\begin{array}{rr} 3863 & 3866 \\ 3594 & 3535 \end{array}\right)$.
The torsion field $K:=\Q(E[6160])$ is a degree-$1634992128000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6160\Z)$.
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 | $2$ | \( 539 = 7^{2} \cdot 11 \) |
| $5$ | split multiplicative | $6$ | \( 94864 = 2^{4} \cdot 7^{2} \cdot 11^{2} \) |
| $7$ | additive | $32$ | \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \) |
| $11$ | additive | $42$ | \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 474320fm
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 16940b2, its twist by $-308$.
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 $0$.