Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-2606649633x-48146388652863\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-2606649633xz^2-48146388652863z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-211138620300x-35099350743798000\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(411504042970655573307/20772802482961, 8347562991967676689193971607100/94676596559838525959)$ | $43.642976026813345327059596242$ | $\infty$ |
| $(-35193, 0)$ | $0$ | $2$ |
| $(58557, 0)$ | $0$ | $2$ |
Integral points
\( \left(-35193, 0\right) \), \( \left(-23363, 0\right) \), \( \left(58557, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 436800 \) | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $132072351989760000000000000000$ | = | $2^{36} \cdot 3^{2} \cdot 5^{16} \cdot 7^{2} \cdot 13^{4} $ |
|
| j-invariant: | $j$ | = | \( \frac{478202393398338853167169}{32244226560000000000} \) | = | $2^{-18} \cdot 3^{-2} \cdot 5^{-10} \cdot 7^{-2} \cdot 13^{-4} \cdot 271^{3} \cdot 288559^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.3365900015036169798922349969$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.4921502744466488284660071481$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.023218976193033$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.902536074718474$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $43.642976026813345327059596242$ |
|
| Real period: | $\Omega$ | ≈ | $0.021218569913632766944515343482$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $7.4083323045115021041093516750 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 7.408332305 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.021219 \cdot 43.642976 \cdot 128}{4^2} \\ & \approx 7.408332305\end{aligned}$$
Modular invariants
Modular form 436800.2.a.ho
For more coefficients, see the Downloads section to the right.
| Modular degree: | 424673280 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{26}^{*}$ | additive | -1 | 6 | 36 | 18 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $4$ | $I_{10}^{*}$ | additive | 1 | 2 | 16 | 10 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{4}$ | 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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 837 & 4 \\ 836 & 5 \end{array}\right),\left(\begin{array}{rr} 419 & 836 \\ 838 & 831 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 241 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 2 \\ 278 & 839 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 627 & 836 \\ 418 & 837 \end{array}\right),\left(\begin{array}{rr} 669 & 838 \\ 674 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$1486356480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 | $2$ | \( 25 = 5^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 145600 = 2^{6} \cdot 5^{2} \cdot 7 \cdot 13 \) |
| $5$ | additive | $14$ | \( 17472 = 2^{6} \cdot 3 \cdot 7 \cdot 13 \) |
| $7$ | split 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.
Its isogeny class 436800ho
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2730z2, its twist by $-40$.
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.