Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+2040299x-923360077\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+2040299xz^2-923360077z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+2644228125x-43088220425250\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(417, -209)$ | $0$ | $2$ |
Integral points
\( \left(417, -209\right) \)
Invariants
| Conductor: | $N$ | = | \( 444675 \) | = | $3 \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-912034013794628484375$ | = | $-1 \cdot 3^{8} \cdot 5^{6} \cdot 7^{3} \cdot 11^{10} $ |
|
| j-invariant: | $j$ | = | \( \frac{98931640625}{96059601} \) | = | $3^{-8} \cdot 5^{9} \cdot 11^{-4} \cdot 37^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7088987000218965102489883334$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.21875457014183272464129869194$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.1307725776312423$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.2444447896841995$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.085810530599303913932368319210$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{3}\cdot2\cdot2\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.7459369791777252458357862147 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.745936979 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.085811 \cdot 1.000000 \cdot 128}{2^2} \\ & \approx 2.745936979\end{aligned}$$
Modular invariants
Modular form 444675.2.a.gp
For more coefficients, see the Downloads section to the right.
| Modular degree: | 22118400 |
|
| $ \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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $11$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 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 |
|---|---|---|
| $2$ | 2B | 4.12.0.11 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 616 = 2^{3} \cdot 7 \cdot 11 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 609 & 8 \\ 608 & 9 \end{array}\right),\left(\begin{array}{rr} 313 & 156 \\ 156 & 309 \end{array}\right),\left(\begin{array}{rr} 313 & 156 \\ 176 & 471 \end{array}\right),\left(\begin{array}{rr} 360 & 3 \\ 221 & 600 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 57 & 8 \\ 228 & 33 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 608 & 595 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[616])$ is a degree-$851558400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/616\Z)$.
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 | $2$ | \( 21175 = 5^{2} \cdot 7 \cdot 11^{2} \) |
| $3$ | split multiplicative | $4$ | \( 148225 = 5^{2} \cdot 7^{2} \cdot 11^{2} \) |
| $5$ | additive | $14$ | \( 17787 = 3 \cdot 7^{2} \cdot 11^{2} \) |
| $7$ | additive | $20$ | \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 444675.gp
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1617.f2, its twist by $-55$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.