Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3+x^2-4653x+118503\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3+x^2z-4653xz^2+118503z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-6030963x+5619337038\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(89, 603)$ | $1.8386156808724258521085493563$ | $\infty$ |
| $(171/4, -171/8)$ | $0$ | $2$ |
Integral points
\( \left(63, 252\right) \), \( \left(63, -315\right) \), \( \left(89, 603\right) \), \( \left(89, -692\right) \)
Invariants
| Conductor: | $N$ | = | \( 451770 \) | = | $2 \cdot 3 \cdot 5 \cdot 11 \cdot 37^{2}$ |
|
| Discriminant: | $\Delta$ | = | $182783883150$ | = | $2 \cdot 3^{8} \cdot 5^{2} \cdot 11 \cdot 37^{3} $ |
|
| j-invariant: | $j$ | = | \( \frac{220020692653}{3608550} \) | = | $2^{-1} \cdot 3^{-8} \cdot 5^{-2} \cdot 11^{-1} \cdot 6037^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.95999914896456585820738820857$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.057269670803509747115364290812$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.8897149085445184$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.83771938627792$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.8386156808724258521085493563$ |
|
| Real period: | $\Omega$ | ≈ | $1.0135129637485875544444429766$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2\cdot2\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $3.7269216558312791329516586729 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 3.726921656 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.013513 \cdot 1.838616 \cdot 8}{2^2} \\ & \approx 3.726921656\end{aligned}$$
Modular invariants
Modular form 451770.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 645120 |
|
| $ \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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $37$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 16280 = 2^{3} \cdot 5 \cdot 11 \cdot 37 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 16277 & 4 \\ 16276 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 8139 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5922 & 1 \\ 13319 & 0 \end{array}\right),\left(\begin{array}{rr} 15844 & 1 \\ 12319 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 3257 & 4 \\ 6514 & 9 \end{array}\right),\left(\begin{array}{rr} 2036 & 14249 \\ 6105 & 10176 \end{array}\right)$.
The torsion field $K:=\Q(E[16280])$ is a degree-$1477799313408000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/16280\Z)$.
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 | $4$ | \( 407 = 11 \cdot 37 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 150590 = 2 \cdot 5 \cdot 11 \cdot 37^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 90354 = 2 \cdot 3 \cdot 11 \cdot 37^{2} \) |
| $11$ | split multiplicative | $12$ | \( 41070 = 2 \cdot 3 \cdot 5 \cdot 37^{2} \) |
| $37$ | additive | $362$ | \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 451770d
consists of 2 curves linked by isogenies of
degree 2.
Twists
This elliptic curve is its own minimal quadratic twist.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.