Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-14850000\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-14850000z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-14850000\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{92718849}{201601}, \frac{821832176007}{90518849}\right) \) | $17.011964598445119562722454717$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([41630763201:821832176007:90518849]\) | $17.011964598445119562722454717$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{92718849}{201601}, \frac{821832176007}{90518849}\right) \) | $17.011964598445119562722454717$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 435600 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-95265720000000000$ | = | $-1 \cdot 2^{12} \cdot 3^{9} \cdot 5^{10} \cdot 11^{2} $ |
|
| j-invariant: | $j$ | = | \( 0 \) | = | $0$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-3})/2]\) (potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9368364411278017324847535606$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.3211174284280379149898691959$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.5850621331742047$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $17.011964598445119562722454717$ |
|
| Real period: | $\Omega$ | ≈ | $0.15490923363873563138801868952$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2\cdot1\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $5.2706207972688688259861922489 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 5.270620797 \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 0.154909 \cdot 17.011965 \cdot 2}{1^2} \\ & \approx 5.270620797\end{aligned}$$
Modular invariants
Modular form 435600.2.a.b
\( q - 5 q^{7} - 7 q^{13} + 8 q^{19} + O(q^{20}) \)
![Copy content]()
![Toggle raw display]()
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5909760 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $5$ | $1$ | $II^{*}$ | additive | 1 | 2 | 10 | 0 |
| $11$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
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 | $2$ | \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \) |
| $3$ | additive | $2$ | \( 48400 = 2^{4} \cdot 5^{2} \cdot 11^{2} \) |
| $5$ | additive | $2$ | \( 17424 = 2^{4} \cdot 3^{2} \cdot 11^{2} \) |
| $11$ | additive | $32$ | \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 435600.b
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 27225.w1, its twist by $60$.
The minimal sextic twist of this elliptic curve is 27.a4, its sextic twist by $-21741885$.
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.