Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-2303x+46648\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-2303xz^2+46648z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-2303x+46648\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-56, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-56:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-56, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-56, 0\right) \)
\([-56:0:1]\)
\( \left(-56, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 454720 \) | = | $2^{6} \cdot 5 \cdot 7^{2} \cdot 29$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-158308494400$ | = | $-1 \cdot 2^{6} \cdot 5^{2} \cdot 7^{6} \cdot 29^{2} $ |
|
| j-invariant: | $j$ | = | \( -\frac{179406144}{21025} \) | = | $-1 \cdot 2^{6} \cdot 3^{3} \cdot 5^{-2} \cdot 29^{-2} \cdot 47^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.88496662044056434270451597096$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.43456204436706496455677646149$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.8723594635381978$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.6884732484474725$ | |||
| 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: | $\Omega$ | ≈ | $0.99520200446520487405158632208$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.9904040089304097481031726442 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 1.990404009 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.995202 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 1.990404009\end{aligned}$$
Modular invariants
Modular form 454720.2.a.bw
For more coefficients, see the Downloads section to the right.
| Modular degree: | 276480 |
|
| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | -1 | 6 | 6 | 0 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $29$ | $2$ | $I_{2}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.6.0.5 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8120 = 2^{3} \cdot 5 \cdot 7 \cdot 29 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 6497 & 3486 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4059 & 5796 \\ 6958 & 3471 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7281 & 3486 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 6089 & 3472 \\ 3913 & 6943 \end{array}\right),\left(\begin{array}{rr} 1159 & 0 \\ 0 & 8119 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 10 & 27 \end{array}\right),\left(\begin{array}{rr} 8113 & 8 \\ 8112 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[8120])$ is a degree-$21121125580800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8120\Z)$.
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 | $2$ | \( 49 = 7^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 90944 = 2^{6} \cdot 7^{2} \cdot 29 \) |
| $7$ | additive | $26$ | \( 9280 = 2^{6} \cdot 5 \cdot 29 \) |
| $29$ | split multiplicative | $30$ | \( 15680 = 2^{6} \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 454720bw
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 4640b2, its twist by $-56$.
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 $0$.