Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+1052532x+324417808\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+1052532xz^2+324417808z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+1052532x+324417808\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(482, 30720)$ | $3.6109503924929210715972413522$ | $\infty$ |
| $(221, 23829)$ | $4.0634065407395391014145842583$ | $\infty$ |
| $(-286, 0)$ | $0$ | $2$ |
Integral points
\( \left(-286, 0\right) \), \((-256,\pm 6180)\), \((221,\pm 23829)\), \((482,\pm 30720)\), \((1404,\pm 67600)\), \((2274,\pm 120320)\), \((42978,\pm 8912384)\)
Invariants
| Conductor: | $N$ | = | \( 486720 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 13^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-120091934173062758400$ | = | $-1 \cdot 2^{24} \cdot 3^{3} \cdot 5^{2} \cdot 13^{9} $ |
|
| j-invariant: | $j$ | = | \( \frac{3774555693}{3515200} \) | = | $2^{-6} \cdot 3^{6} \cdot 5^{-2} \cdot 13^{-3} \cdot 173^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5401232073675051720447338971$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.056725314370208582956669315101$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.949724584751915$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.06353000370528$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
|
| Mordell-Weil rank: | $r$ | = | $ 2$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $12.880339330240526535047988069$ |
|
| Real period: | $\Omega$ | ≈ | $0.12197095830832357479748156451$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $12.568218651566621270468658097 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 12.568218652 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.121971 \cdot 12.880339 \cdot 32}{2^2} \\ & \approx 12.568218652\end{aligned}$$
Modular invariants
Modular form 486720.2.a.p
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18579456 |
|
| $ \Gamma_0(N) $-optimal: | not computed* (one of 4 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{14}^{*}$ | additive | -1 | 6 | 24 | 6 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
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 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 779 & 1548 \\ 1554 & 1487 \end{array}\right),\left(\begin{array}{rr} 937 & 12 \\ 942 & 73 \end{array}\right),\left(\begin{array}{rr} 272 & 11 \\ 1005 & 1528 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 110 & 1557 \\ 1107 & 8 \end{array}\right),\left(\begin{array}{rr} 321 & 62 \\ 1474 & 1219 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 1510 & 1551 \end{array}\right),\left(\begin{array}{rr} 1549 & 12 \\ 1548 & 13 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$9661317120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $2$ | \( 507 = 3 \cdot 13^{2} \) |
| $3$ | additive | $6$ | \( 54080 = 2^{6} \cdot 5 \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 97344 = 2^{6} \cdot 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 486720.p
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 1170.c4, its twist by $312$.
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.