Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-46200x-1541000\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-46200xz^2-1541000z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-46200x-1541000\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-94, 1404\right) \) | $3.6375610682231074143609793360$ | $\infty$ |
| \( \left(230, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-94:1404:1]\) | $3.6375610682231074143609793360$ | $\infty$ |
| \([230:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-94, 1404\right) \) | $3.6375610682231074143609793360$ | $\infty$ |
| \( \left(230, 0\right) \) | $0$ | $2$ |
Integral points
\((-94,\pm 1404)\), \( \left(230, 0\right) \)
\([-94:\pm 1404:1]\), \([230:0:1]\)
\((-94,\pm 1404)\), \( \left(230, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 417600 \) | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 29$ |
|
| Minimal Discriminant: | $\Delta$ | = | $5285250000000000$ | = | $2^{10} \cdot 3^{6} \cdot 5^{12} \cdot 29 $ |
|
| j-invariant: | $j$ | = | \( \frac{934979584}{453125} \) | = | $2^{11} \cdot 5^{-6} \cdot 7^{3} \cdot 11^{3} \cdot 29^{-1}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7105222674191645607967421661$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.22112548359856156338228688686$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.894034150371482$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.3870238151376495$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.6375610682231074143609793360$ |
|
| Real period: | $\Omega$ | ≈ | $0.34186074239218439192769147034$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2^{2}\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.9741573091186351914402762437 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.974157309 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.341861 \cdot 3.637561 \cdot 16}{2^2} \\ & \approx 4.974157309\end{aligned}$$
Modular invariants
Modular form 417600.2.a.z
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1769472 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | -1 | 6 | 10 | 0 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $4$ | $I_{6}^{*}$ | additive | 1 | 2 | 12 | 6 |
| $29$ | $1$ | $I_{1}$ | nonsplit 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 580 = 2^{2} \cdot 5 \cdot 29 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 577 & 4 \\ 576 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 437 & 146 \\ 144 & 435 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 62 & 1 \\ 259 & 0 \end{array}\right),\left(\begin{array}{rr} 117 & 4 \\ 234 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[580])$ is a degree-$2619187200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/580\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \) |
| $3$ | additive | $6$ | \( 46400 = 2^{6} \cdot 5^{2} \cdot 29 \) |
| $5$ | additive | $18$ | \( 16704 = 2^{6} \cdot 3^{2} \cdot 29 \) |
| $29$ | nonsplit multiplicative | $30$ | \( 14400 = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 417600.z
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1160.c2, its twist by $120$.
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.