Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-5007468x+4312959568\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-5007468xz^2+4312959568z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-5007468x+4312959568\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1461, 10751\right) \) | $5.8926284942480831537820536679$ | $\infty$ |
| \( \left(1292, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1461:10751:1]\) | $5.8926284942480831537820536679$ | $\infty$ |
| \([1292:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1461, 10751\right) \) | $5.8926284942480831537820536679$ | $\infty$ |
| \( \left(1292, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(1292, 0\right) \), \((1461,\pm 10751)\)
\([1292:0:1]\), \([1461:\pm 10751:1]\)
\( \left(1292, 0\right) \), \((1461,\pm 10751)\)
Invariants
| Conductor: | $N$ | = | \( 469440 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 163$ |
|
| Minimal Discriminant: | $\Delta$ | = | $25231325921280$ | = | $2^{19} \cdot 3^{10} \cdot 5 \cdot 163 $ |
|
| j-invariant: | $j$ | = | \( \frac{72662579271908569}{132030} \) | = | $2^{-1} \cdot 3^{-4} \cdot 5^{-1} \cdot 23^{3} \cdot 163^{-1} \cdot 18143^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2569276493714902662647993193$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.66790073419751745644132851865$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.950446983681939$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.433081760543789$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.8926284942480831537820536679$ |
|
| Real period: | $\Omega$ | ≈ | $0.43423270890369869290538407157$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2^{2}\cdot1\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $5.1175440672409364794099517079 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 5.117544067 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.434233 \cdot 5.892628 \cdot 8}{2^2} \\ & \approx 5.117544067\end{aligned}$$
Modular invariants
Modular form 469440.2.a.e
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6291456 |
|
| $ \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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{9}^{*}$ | additive | -1 | 6 | 19 | 1 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $163$ | $1$ | $I_{1}$ | 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 | 8.12.0.12 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 19560 = 2^{3} \cdot 3 \cdot 5 \cdot 163 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 19553 & 8 \\ 19552 & 9 \end{array}\right),\left(\begin{array}{rr} 17112 & 7327 \\ 2453 & 2466 \end{array}\right),\left(\begin{array}{rr} 13039 & 19552 \\ 13036 & 19527 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 10448 & 3 \\ 4085 & 2 \end{array}\right),\left(\begin{array}{rr} 12224 & 2437 \\ 12221 & 12192 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 19554 & 19555 \end{array}\right),\left(\begin{array}{rr} 7828 & 1 \\ 15671 & 6 \end{array}\right)$.
The torsion field $K:=\Q(E[19560])$ is a degree-$517242181386240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/19560\Z)$.
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 | $4$ | \( 7335 = 3^{2} \cdot 5 \cdot 163 \) |
| $3$ | additive | $8$ | \( 52160 = 2^{6} \cdot 5 \cdot 163 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 93888 = 2^{6} \cdot 3^{2} \cdot 163 \) |
| $163$ | nonsplit multiplicative | $164$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 469440e
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 4890b3, its twist by $24$.
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.