Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-4051308x+3077600848\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-4051308xz^2+3077600848z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-4051308x+3077600848\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{176711657904}{91107025}, \frac{43624858178708708}{869616553625}\right) \) | $25.124273693469203624936128054$ | $\infty$ |
| \( \left(1292, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1686712774693680:43624858178708708:869616553625]\) | $25.124273693469203624936128054$ | $\infty$ |
| \([1292:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{176711657904}{91107025}, \frac{43624858178708708}{869616553625}\right) \) | $25.124273693469203624936128054$ | $\infty$ |
| \( \left(1292, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(1292, 0\right) \)
\([1292:0:1]\)
\( \left(1292, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 469440 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 163$ |
|
| Minimal Discriminant: | $\Delta$ | = | $163905733559408394240$ | = | $2^{18} \cdot 3^{11} \cdot 5 \cdot 163^{4} $ |
|
| j-invariant: | $j$ | = | \( \frac{38480618749557529}{857682789615} \) | = | $3^{-5} \cdot 5^{-1} \cdot 163^{-4} \cdot 337609^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6662728754609689201940485348$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0772459602869961103705777342$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9476298241848735$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.384405949110108$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $25.124273693469203624936128054$ |
|
| Real period: | $\Omega$ | ≈ | $0.18137394277601317015771989791$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $9.1137771583363527876691958554 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 9.113777158 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.181374 \cdot 25.124274 \cdot 8}{2^2} \\ & \approx 9.113777158\end{aligned}$$
Modular invariants
Modular form 469440.2.a.bn
For more coefficients, see the Downloads section to the right.
| Modular degree: | 15728640 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{8}^{*}$ | additive | -1 | 6 | 18 | 0 |
| $3$ | $2$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $163$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 | 8.12.0.8 | $12$ |
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
$\left(\begin{array}{rr} 19553 & 8 \\ 19552 & 9 \end{array}\right),\left(\begin{array}{rr} 13036 & 19559 \\ 6497 & 19554 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 19554 & 19555 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 8641 & 8 \\ 15004 & 33 \end{array}\right),\left(\begin{array}{rr} 12221 & 12222 \\ 17102 & 7325 \end{array}\right),\left(\begin{array}{rr} 17112 & 7327 \\ 17087 & 17040 \end{array}\right),\left(\begin{array}{rr} 15656 & 3 \\ 5 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
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 | $2$ | \( 45 = 3^{2} \cdot 5 \) |
| $3$ | 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 469440.bn
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2445.c1, 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.