Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-86851x+7977098\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-86851xz^2+7977098z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-112558275x+372517170750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-108, 4066)$ | $4.3080845690903415251702532808$ | $\infty$ |
| $(-333, 166)$ | $0$ | $2$ |
Integral points
\( \left(-333, 166\right) \), \( \left(-108, 4066\right) \), \( \left(-108, -3959\right) \)
Invariants
| Conductor: | $N$ | = | \( 476850 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 11 \cdot 17^{2}$ |
|
| Discriminant: | $\Delta$ | = | $14387499722062500$ | = | $2^{2} \cdot 3 \cdot 5^{6} \cdot 11 \cdot 17^{8} $ |
|
| j-invariant: | $j$ | = | \( \frac{192100033}{38148} \) | = | $2^{-2} \cdot 3^{-1} \cdot 11^{-1} \cdot 17^{-2} \cdot 577^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8167402988293288138199956806$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.40458532941582941360515129495$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.8399725522135814$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.497482511535867$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.3080845690903415251702532808$ |
|
| Real period: | $\Omega$ | ≈ | $0.37485918625984444622871199433$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot1\cdot2\cdot1\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $6.4597003036311921349962351494 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 6.459700304 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.374859 \cdot 4.308085 \cdot 16}{2^2} \\ & \approx 6.459700304\end{aligned}$$
Modular invariants
Modular form 476850.2.a.ds
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3538944 |
|
| $ \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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $17$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4488 = 2^{3} \cdot 3 \cdot 11 \cdot 17 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1498 & 1 \\ 1495 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1057 & 4 \\ 2114 & 9 \end{array}\right),\left(\begin{array}{rr} 2245 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 3929 & 562 \\ 560 & 3927 \end{array}\right),\left(\begin{array}{rr} 818 & 1 \\ 4079 & 0 \end{array}\right),\left(\begin{array}{rr} 4485 & 4 \\ 4484 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[4488])$ is a degree-$6353112268800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4488\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 238425 = 3 \cdot 5^{2} \cdot 11 \cdot 17^{2} \) |
| $3$ | split multiplicative | $4$ | \( 158950 = 2 \cdot 5^{2} \cdot 11 \cdot 17^{2} \) |
| $5$ | additive | $14$ | \( 19074 = 2 \cdot 3 \cdot 11 \cdot 17^{2} \) |
| $11$ | split multiplicative | $12$ | \( 43350 = 2 \cdot 3 \cdot 5^{2} \cdot 17^{2} \) |
| $17$ | additive | $162$ | \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 476850.ds
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1122.l2, its twist by $85$.
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.