Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2+7704x-30864\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z+7704xz^2-30864z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+623997x-20627838\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(68, 896\right) \) | $2.7913349382466442533088691733$ | $\infty$ |
| \( \left(4, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([68:896:1]\) | $2.7913349382466442533088691733$ | $\infty$ |
| \([4:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(609, 24192\right) \) | $2.7913349382466442533088691733$ | $\infty$ |
| \( \left(33, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(4, 0\right) \), \((68,\pm 896)\), \((125,\pm 1694)\)
\([4:0:1]\), \([68:\pm 896:1]\), \([125:\pm 1694:1]\)
\( \left(4, 0\right) \), \((68,\pm 896)\), \((125,\pm 1694)\)
Invariants
| Conductor: | $N$ | = | \( 493680 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 17$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-29605760532480$ | = | $-1 \cdot 2^{16} \cdot 3 \cdot 5 \cdot 11^{6} \cdot 17 $ |
|
| j-invariant: | $j$ | = | \( \frac{6967871}{4080} \) | = | $2^{-4} \cdot 3^{-1} \cdot 5^{-1} \cdot 17^{-1} \cdot 191^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2743220772662725090153920325$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.61777273969285807243281187794$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9196558505806328$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.9338677366193378$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.7913349382466442533088691733$ |
|
| Real period: | $\Omega$ | ≈ | $0.38983059141671094779356163630$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot1\cdot1\cdot2^{2}\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.3525909992752706430407685711 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.352590999 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.389831 \cdot 2.791335 \cdot 16}{2^2} \\ & \approx 4.352590999\end{aligned}$$
Modular invariants
Modular form 493680.2.a.bb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 983040 |
|
| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{8}^{*}$ | additive | -1 | 4 | 16 | 4 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $17$ | $1$ | $I_{1}$ | 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 | 4.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 22440 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 22433 & 8 \\ 22432 & 9 \end{array}\right),\left(\begin{array}{rr} 21088 & 12243 \\ 12925 & 4082 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 3004 & 4081 \\ 1463 & 20406 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7391 & 7392 \\ 242 & 5345 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 22434 & 22435 \end{array}\right),\left(\begin{array}{rr} 10616 & 12243 \\ 20405 & 4082 \end{array}\right),\left(\begin{array}{rr} 4079 & 0 \\ 0 & 22439 \end{array}\right),\left(\begin{array}{rr} 3323 & 9438 \\ 770 & 18107 \end{array}\right)$.
The torsion field $K:=\Q(E[22440])$ is a degree-$762373472256000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/22440\Z)$.
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 | $2$ | \( 30855 = 3 \cdot 5 \cdot 11^{2} \cdot 17 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 164560 = 2^{4} \cdot 5 \cdot 11^{2} \cdot 17 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 98736 = 2^{4} \cdot 3 \cdot 11^{2} \cdot 17 \) |
| $11$ | additive | $62$ | \( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 29040 = 2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 493680bb
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 510f1, its twist by $44$.
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.