Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+x^2-41184x+3199767\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+x^2z-41184xz^2+3199767z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-53374491x+150088954806\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(467/4, -471/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 858 \) | = | $2 \cdot 3 \cdot 11 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $7722$ | = | $2 \cdot 3^{3} \cdot 11 \cdot 13 $ |
|
| j-invariant: | $j$ | = | \( \frac{7725203825376001537}{7722} \) | = | $2^{-1} \cdot 3^{-3} \cdot 11^{-1} \cdot 13^{-1} \cdot 479^{3} \cdot 4127^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.93426579546086537571547285165$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.93426579546086537571547285165$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.059433311036222$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.438722276082378$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $1.8422546597096107137513351422$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.8422546597096107137513351422 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
|
BSD formula
$$\begin{aligned} 1.842254660 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 1.842255 \cdot 1.000000 \cdot 1}{2^2} \\ & \approx 1.842254660\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1152 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | 8.12.0.6 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 2296 & 3 \\ 2293 & 2 \end{array}\right),\left(\begin{array}{rr} 536 & 3 \\ 2909 & 2 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3426 & 3427 \end{array}\right),\left(\begin{array}{rr} 3425 & 8 \\ 3424 & 9 \end{array}\right),\left(\begin{array}{rr} 432 & 2153 \\ 2995 & 2982 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 3011 & 3004 \\ 3050 & 1293 \end{array}\right),\left(\begin{array}{rr} 316 & 1 \\ 959 & 6 \end{array}\right)$.
The torsion field $K:=\Q(E[3432])$ is a degree-$531372441600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3432\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 429 = 3 \cdot 11 \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 286 = 2 \cdot 11 \cdot 13 \) |
| $11$ | split multiplicative | $12$ | \( 78 = 2 \cdot 3 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 66 = 2 \cdot 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 858h
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
This elliptic curve is its own minimal quadratic twist.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{858}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{22}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{39}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{22}, \sqrt{39})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.4.106309632.4 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.14632310742192.4 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 11 | 13 |
|---|---|---|---|---|
| Reduction type | split | nonsplit | split | split |
| $\lambda$-invariant(s) | 1 | 0 | 3 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.