Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2+56170138367x-132426257594304863\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z+56170138367xz^2-132426257594304863z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+4549781207700x-96538728136904622000\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(473047, 0)$ | $0$ | $2$ |
Integral points
\( \left(473047, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 436800 \) | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $-7587200354233999679631360000000000000$ | = | $-1 \cdot 2^{24} \cdot 3^{32} \cdot 5^{13} \cdot 7 \cdot 13^{4} $ |
|
| j-invariant: | $j$ | = | \( \frac{4784981304203817469820354951}{1852343836482910078035000000} \) | = | $2^{-6} \cdot 3^{-32} \cdot 5^{-7} \cdot 7^{-1} \cdot 13^{-4} \cdot 1685104151^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $5.7557396227537854021212440113$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.9112998956968172506950161625$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0927873585724972$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.11258250832939$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.0034767361261480435798017293616$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.3906944504592174319206917446 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $100$ = $10^2$ (exact) |
|
BSD formula
$$\begin{aligned} 1.390694450 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{100 \cdot 0.003477 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 1.390694450\end{aligned}$$
Modular invariants
Modular form 436800.2.a.u
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7927234560 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{14}^{*}$ | additive | -1 | 6 | 24 | 6 |
| $3$ | $2$ | $I_{32}$ | nonsplit multiplicative | 1 | 1 | 32 | 32 |
| $5$ | $2$ | $I_{7}^{*}$ | additive | 1 | 2 | 13 | 7 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $2$ | $I_{4}$ | nonsplit 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 | 8.12.0.8 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 2904 & 3637 \\ 3635 & 3638 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3634 & 3635 \end{array}\right),\left(\begin{array}{rr} 561 & 8 \\ 2244 & 33 \end{array}\right),\left(\begin{array}{rr} 1357 & 1364 \\ 1318 & 3179 \end{array}\right),\left(\begin{array}{rr} 3633 & 8 \\ 3632 & 9 \end{array}\right),\left(\begin{array}{rr} 2084 & 1 \\ 1063 & 6 \end{array}\right),\left(\begin{array}{rr} 3181 & 3182 \\ 2262 & 445 \end{array}\right)$.
The torsion field $K:=\Q(E[3640])$ is a degree-$811550638080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3640\Z)$.
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 | $2$ | \( 175 = 5^{2} \cdot 7 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 145600 = 2^{6} \cdot 5^{2} \cdot 7 \cdot 13 \) |
| $5$ | additive | $18$ | \( 17472 = 2^{6} \cdot 3 \cdot 7 \cdot 13 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 62400 = 2^{6} \cdot 3 \cdot 5^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 33600 = 2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 436800u
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2730k4, its twist by $-40$.
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 $0$.