Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2-707850x-222715625\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z-707850xz^2-222715625z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-11325603x-14265125602\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-550, 275)$ | $0$ | $2$ |
| $(-418, 209)$ | $0$ | $2$ |
Integral points
\( \left(-550, 275\right) \), \( \left(-418, 209\right) \)
Invariants
| Conductor: | $N$ | = | \( 21285 \) | = | $3^{2} \cdot 5 \cdot 11 \cdot 43$ |
|
| Discriminant: | $\Delta$ | = | $1236621452012780625$ | = | $3^{14} \cdot 5^{4} \cdot 11^{2} \cdot 43^{4} $ |
|
| j-invariant: | $j$ | = | \( \frac{53804702959424445601}{1696325722925625} \) | = | $3^{-8} \cdot 5^{-4} \cdot 11^{-2} \cdot 43^{-4} \cdot 3775201^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2463838334638934357773138626$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.6970776891298385900796912441$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9586366296458889$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.220231809798175$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.16491215020635370149416585379$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.9684187037143666268949853682 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
|
BSD formula
$$\begin{aligned} 2.968418704 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.164912 \cdot 1.000000 \cdot 32}{4^2} \\ & \approx 2.968418704\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 327680 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $4$ | $I_{8}^{*}$ | additive | -1 | 2 | 14 | 8 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $43$ | $2$ | $I_{4}$ | 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 | 4.12.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 56760 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 43 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 7 & 42576 \\ 56754 & 14185 \end{array}\right),\left(\begin{array}{rr} 34057 & 37848 \\ 41628 & 37873 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 47305 & 23652 \\ 4752 & 9469 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 56756 & 56757 \end{array}\right),\left(\begin{array}{rr} 30361 & 18924 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 36127 & 6 \\ 25794 & 56755 \end{array}\right),\left(\begin{array}{rr} 18919 & 0 \\ 0 & 56759 \end{array}\right),\left(\begin{array}{rr} 56753 & 8 \\ 56752 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[56760])$ is a degree-$8120188403712000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/56760\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 9 = 3^{2} \) |
| $3$ | additive | $8$ | \( 2365 = 5 \cdot 11 \cdot 43 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 4257 = 3^{2} \cdot 11 \cdot 43 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 1935 = 3^{2} \cdot 5 \cdot 43 \) |
| $43$ | nonsplit multiplicative | $44$ | \( 495 = 3^{2} \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 21285.e
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 7095.c3, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{11})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-11})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.303595776.1 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 5 | 11 | 43 |
|---|---|---|---|---|---|
| Reduction type | ord | add | nonsplit | nonsplit | nonsplit |
| $\lambda$-invariant(s) | 3 | - | 0 | 0 | 0 |
| $\mu$-invariant(s) | 0 | - | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.