Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-779001x+253003648\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-779001xz^2+253003648z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-1009584675x+11807166966750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-103, 18276)$ | $0.26855259110033269204285804321$ | $\infty$ |
| $(-4057/4, 4053/8)$ | $0$ | $2$ |
Integral points
\( \left(-103, 18276\right) \), \( \left(-103, -18174\right) \), \( \left(347, 4776\right) \), \( \left(347, -5124\right) \), \( \left(392, 2616\right) \), \( \left(392, -3009\right) \), \( \left(626, 2967\right) \), \( \left(626, -3594\right) \), \( \left(1517, 49866\right) \), \( \left(1517, -51384\right) \), \( \left(2408, 109887\right) \), \( \left(2408, -112296\right) \), \( \left(29017, 4926116\right) \), \( \left(29017, -4955134\right) \)
Invariants
| Conductor: | $N$ | = | \( 2850 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 19$ |
|
| Discriminant: | $\Delta$ | = | $2587847797617187500$ | = | $2^{2} \cdot 3^{20} \cdot 5^{10} \cdot 19 $ |
|
| j-invariant: | $j$ | = | \( \frac{3345930611358906241}{165622259047500} \) | = | $2^{-2} \cdot 3^{-20} \cdot 5^{-4} \cdot 11^{6} \cdot 19^{-1} \cdot 47^{3} \cdot 263^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2922073441374617107045119454$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4874883879204115234041322788$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.081271340032757$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.575790774056907$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.26855259110033269204285804321$ |
|
| Real period: | $\Omega$ | ≈ | $0.25335331746360284608330836187$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 160 $ = $ 2\cdot( 2^{2} \cdot 5 )\cdot2^{2}\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $2.7215475947486285155903510170 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 2.721547595 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.253353 \cdot 0.268553 \cdot 160}{2^2} \\ & \approx 2.721547595\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 92160 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $20$ | $I_{20}$ | split multiplicative | -1 | 1 | 20 | 20 |
| $5$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $19$ | $1$ | $I_{1}$ | 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 | 4.6.0.1 |
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 $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1688 & 3 \\ 485 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 859 & 858 \\ 298 & 1435 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 288 & 1433 \\ 313 & 360 \end{array}\right),\left(\begin{array}{rr} 761 & 8 \\ 764 & 33 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 2274 & 2275 \end{array}\right),\left(\begin{array}{rr} 2273 & 8 \\ 2272 & 9 \end{array}\right),\left(\begin{array}{rr} 1823 & 2272 \\ 452 & 2247 \end{array}\right)$.
The torsion field $K:=\Q(E[2280])$ is a degree-$90773913600$ 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 | $4$ | \( 475 = 5^{2} \cdot 19 \) |
| $3$ | split multiplicative | $4$ | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
| $5$ | additive | $18$ | \( 38 = 2 \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 2850l
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 570i3, its twist by $5$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{19}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-95}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{19})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.120437455360000.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1197711360000.6 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.5771493546750000.10 | \(\Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | split | add | ord | ss | ord | ord | split | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 2 | 2 | - | 1 | 3,1 | 3 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | 1 | 1,1 |
| $\mu$-invariant(s) | 1 | 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.