Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-274776x-55916177\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-274776xz^2-55916177z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-356109075x-2607756815250\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{2423}{4}, -\frac{2427}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([4846:-2427:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(21810, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 489675 \) | = | $3 \cdot 5^{2} \cdot 6529$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-21850098812578125$ | = | $-1 \cdot 3^{8} \cdot 5^{7} \cdot 6529^{2} $ |
|
| j-invariant: | $j$ | = | \( -\frac{146837778453361009}{1398406324005} \) | = | $-1 \cdot 3^{-8} \cdot 5^{-1} \cdot 7^{3} \cdot 6529^{-2} \cdot 75367^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9562015308028610199196565390$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1514825745858108326192768724$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.8761666339678333$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.7553765217123782$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.10420255601515381203104383389$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{3}\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $0.83362044812123049624835067112 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 0.833620448 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.104203 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 0.833620448\end{aligned}$$
Modular invariants
Modular form 489675.2.a.r
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4220928 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $5$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $6529$ | $2$ | $I_{2}$ | nonsplit 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 130580 = 2^{2} \cdot 5 \cdot 6529 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 97936 & 32649 \\ 32645 & 97936 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 52234 & 1 \\ 104463 & 0 \end{array}\right),\left(\begin{array}{rr} 130577 & 4 \\ 130576 & 5 \end{array}\right),\left(\begin{array}{rr} 39181 & 4 \\ 78362 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[130580])$ is a degree-$6976721159140147200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/130580\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 25 = 5^{2} \) |
| $3$ | split multiplicative | $4$ | \( 163225 = 5^{2} \cdot 6529 \) |
| $5$ | additive | $18$ | \( 19587 = 3 \cdot 6529 \) |
| $6529$ | nonsplit multiplicative | $6530$ | \( 75 = 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 489675.r
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 97935.g2, its twist by $5$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.