Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2-557756x-160461300\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z-557756xz^2-160461300z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-45178263x-116840752938\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-434, 216)$ | $3.0742328057655983017904290490$ | $\infty$ |
| $(1162, 27600)$ | $4.2471345401018901985297436375$ | $\infty$ |
| $(-438, 0)$ | $0$ | $2$ |
| $(862, 0)$ | $0$ | $2$ |
Integral points
\( \left(-438, 0\right) \), \((-434,\pm 216)\), \( \left(-425, 0\right) \), \( \left(862, 0\right) \), \((1162,\pm 27600)\), \((1434,\pm 44616)\), \((3787,\pm 228150)\), \((6439,\pm 513084)\)
Invariants
| Conductor: | $N$ | = | \( 446160 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 13^{2}$ |
|
| Discriminant: | $\Delta$ | = | $7569208801440000$ | = | $2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 11^{2} \cdot 13^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{15529488955216}{6125625} \) | = | $2^{4} \cdot 3^{-4} \cdot 5^{-4} \cdot 11^{-2} \cdot 9901^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0112892192269199402438114087$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.26671642012285469927224627361$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0278304524202422$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9442595265428975$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
|
| Mordell-Weil rank: | $r$ | = | $ 2$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $12.806510366044661136668420324$ |
|
| Real period: | $\Omega$ | ≈ | $0.17470233439569719291402253569$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2\cdot2^{2}\cdot2\cdot2\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $17.898618051285574813466157209 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 17.898618051 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.174702 \cdot 12.806510 \cdot 128}{4^2} \\ & \approx 17.898618051\end{aligned}$$
Modular invariants
Modular form 446160.2.a.fn
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4718592 |
|
| $ \Gamma_0(N) $-optimal: | not computed* (one of 4 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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | 1 | 4 | 8 | 0 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 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 |
|---|---|---|
| $2$ | 2Cs | 8.24.0.15 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5720 = 2^{3} \cdot 5 \cdot 11 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 4835 & 3848 \\ 858 & 5381 \end{array}\right),\left(\begin{array}{rr} 4447 & 442 \\ 4342 & 4395 \end{array}\right),\left(\begin{array}{rr} 3745 & 5616 \\ 4264 & 1509 \end{array}\right),\left(\begin{array}{rr} 3079 & 0 \\ 0 & 5719 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5713 & 8 \\ 5712 & 9 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 5716 & 5717 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4577 & 1768 \\ 2028 & 1353 \end{array}\right)$.
The torsion field $K:=\Q(E[5720])$ is a degree-$1328431104000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5720\Z)$.
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 | $2$ | \( 169 = 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 148720 = 2^{4} \cdot 5 \cdot 11 \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 89232 = 2^{4} \cdot 3 \cdot 11 \cdot 13^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 40560 = 2^{4} \cdot 3 \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 446160.fn
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 1320.f4, its twist by $-52$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.