Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2-1682442x+811529716\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z-1682442xz^2+811529716z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-26919075x+51910982750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(989, 10193)$ | $0.55944423902910734998278707294$ | $\infty$ |
| $(2531/4, -2531/8)$ | $0$ | $2$ |
Integral points
\( \left(-721, 40973\right) \), \( \left(-721, -40252\right) \), \( \left(39, 27293\right) \), \( \left(39, -27332\right) \), \( \left(989, 10193\right) \), \( \left(989, -11182\right) \), \( \left(1445, 36641\right) \), \( \left(1445, -38086\right) \)
Invariants
| Conductor: | $N$ | = | \( 8550 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 19$ |
|
| Discriminant: | $\Delta$ | = | $20577868349400000000$ | = | $2^{9} \cdot 3^{7} \cdot 5^{8} \cdot 19^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{46237740924063961}{1806561830400} \) | = | $2^{-9} \cdot 3^{-1} \cdot 5^{-2} \cdot 17^{3} \cdot 19^{-6} \cdot 43^{3} \cdot 491^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4731245127131986303285360435$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1190994121620935973305337584$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0022133277172731$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.032998115808154$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.55944423902910734998278707294$ |
|
| Real period: | $\Omega$ | ≈ | $0.21402896621678819172274191387$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 1\cdot2^{2}\cdot2^{2}\cdot( 2 \cdot 3 ) $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $2.8736945312481022792473987310 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 2.873694531 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.214029 \cdot 0.559444 \cdot 96}{2^2} \\ & \approx 2.873694531\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 165888 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
| $3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $19$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
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 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 11 & 2 \\ 2230 & 2271 \end{array}\right),\left(\begin{array}{rr} 1823 & 2268 \\ 1818 & 2207 \end{array}\right),\left(\begin{array}{rr} 10 & 3 \\ 1113 & 2272 \end{array}\right),\left(\begin{array}{rr} 2269 & 12 \\ 2268 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 370 & 2277 \\ 1547 & 8 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 666 & 1057 \\ 1805 & 2186 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1921 & 12 \\ 126 & 73 \end{array}\right)$.
The torsion field $K:=\Q(E[2280])$ is a degree-$45386956800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2280\Z)$.
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 | $4$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $8$ | \( 25 = 5^{2} \) |
| $5$ | additive | $18$ | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 8550l
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 570f4, its twist by $-15$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.866400.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{6})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.553584375.1 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.432373800960000.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.750648960000.7 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.556276750938120501658868496690673828125.3 | \(\Z/18\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | add | add | ord | ss | ord | ss | split | ss | ord | ord | ord | ss | ord | ss |
| $\lambda$-invariant(s) | 3 | - | - | 1 | 1,1 | 1 | 1,1 | 2 | 1,1 | 1 | 1 | 1 | 1,1 | 1 | 1,1 |
| $\mu$-invariant(s) | 0 | - | - | 0 | 0,0 | 0 | 0,0 | 0 | 0,0 | 0 | 0 | 0 | 0,0 | 0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.