Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+x^2-490065x-132112203\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+x^2z-490065xz^2-132112203z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-635124267x-6154300070874\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-81385/196, 631737/2744)$ | $10.364769406730435379352725362$ | $\infty$ |
| $(-1661/4, 1657/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 465690 \) | = | $2 \cdot 3 \cdot 5 \cdot 19^{2} \cdot 43$ |
|
| Discriminant: | $\Delta$ | = | $15853532740850250$ | = | $2 \cdot 3^{6} \cdot 5^{3} \cdot 19^{6} \cdot 43^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{276670733768281}{336980250} \) | = | $2^{-1} \cdot 3^{-6} \cdot 5^{-3} \cdot 17^{3} \cdot 43^{-2} \cdot 3833^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0186874559018322139286888256$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.54646796631861198392417510966$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9535732176190836$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9015714039745046$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $10.364769406730435379352725362$ |
|
| Real period: | $\Omega$ | ≈ | $0.18045457235564288717716281833$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 1\cdot2\cdot3\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $11.222220185138346914939368325 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 11.222220185 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.180455 \cdot 10.364769 \cdot 24}{2^2} \\ & \approx 11.222220185\end{aligned}$$
Modular invariants
Modular form 465690.2.a.ce
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6842880 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $19$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $43$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 \( 98040 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \cdot 43 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 61919 & 0 \\ 0 & 98039 \end{array}\right),\left(\begin{array}{rr} 67090 & 87723 \\ 14421 & 30952 \end{array}\right),\left(\begin{array}{rr} 4561 & 56772 \\ 6726 & 46513 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 97990 & 98031 \end{array}\right),\left(\begin{array}{rr} 67090 & 87723 \\ 43833 & 30952 \end{array}\right),\left(\begin{array}{rr} 70529 & 25802 \\ 67944 & 1 \end{array}\right),\left(\begin{array}{rr} 22155 & 59128 \\ 47918 & 2357 \end{array}\right),\left(\begin{array}{rr} 98029 & 12 \\ 98028 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[98040])$ is a degree-$151478423676518400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/98040\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 1805 = 5 \cdot 19^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 31046 = 2 \cdot 19^{2} \cdot 43 \) |
| $5$ | split multiplicative | $6$ | \( 93138 = 2 \cdot 3 \cdot 19^{2} \cdot 43 \) |
| $19$ | additive | $182$ | \( 1290 = 2 \cdot 3 \cdot 5 \cdot 43 \) |
| $43$ | split multiplicative | $44$ | \( 10830 = 2 \cdot 3 \cdot 5 \cdot 19^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 465690.ce
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 1290.h3, its twist by $-19$.
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.