Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-x^2-11168807x-27952935469\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-x^2z-11168807xz^2-27952935469z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-178700907x-1789166570906\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{1924156411}{277729}, \frac{69254881720592}{146363183}\right) \) | $21.312817300230980096954767869$ | $\infty$ |
| \( \left(\frac{16875}{4}, -\frac{16879}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1014030428597:69254881720592:146363183]\) | $21.312817300230980096954767869$ | $\infty$ |
| \([33750:-16879:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{7696347915}{277729}, \frac{558095760931856}{146363183}\right) \) | $21.312817300230980096954767869$ | $\infty$ |
| \( \left(16874, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 446490 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 41$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-248451602804833539424500$ | = | $-1 \cdot 2^{2} \cdot 3^{10} \cdot 5^{3} \cdot 11^{6} \cdot 41^{6} $ |
|
| j-invariant: | $j$ | = | \( -\frac{119305480789133569}{192379221760500} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-4} \cdot 5^{-3} \cdot 7^{3} \cdot 41^{-6} \cdot 70327^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1802130896585429530395092453$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4319593089253028353109148379$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.002573457844343$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.737497519514777$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $21.312817300230980096954767869$ |
|
| Real period: | $\Omega$ | ≈ | $0.039102218230219689971883191606$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2\cdot2^{2}\cdot3\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $20.001082396186402209392429648 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 20.001082396 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.039102 \cdot 21.312817 \cdot 96}{2^2} \\ & \approx 20.001082396\end{aligned}$$
Modular invariants
Modular form 446490.2.a.gb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 59719680 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $11$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $41$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
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 | $3$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 27060 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 41 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 27049 & 12 \\ 27048 & 13 \end{array}\right),\left(\begin{array}{rr} 15786 & 5137 \\ 15785 & 24806 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 27010 & 27051 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 21121 & 7392 \\ 8646 & 17293 \end{array}\right),\left(\begin{array}{rr} 12310 & 22143 \\ 11781 & 14752 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7379 & 0 \\ 0 & 27059 \end{array}\right),\left(\begin{array}{rr} 9019 & 19668 \\ 9020 & 27059 \end{array}\right)$.
The torsion field $K:=\Q(E[27060])$ is a degree-$837933465600000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/27060\Z)$.
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 | $4$ | \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \) |
| $3$ | additive | $8$ | \( 242 = 2 \cdot 11^{2} \) |
| $5$ | split multiplicative | $6$ | \( 89298 = 2 \cdot 3^{2} \cdot 11^{2} \cdot 41 \) |
| $11$ | additive | $62$ | \( 3690 = 2 \cdot 3^{2} \cdot 5 \cdot 41 \) |
| $41$ | nonsplit multiplicative | $42$ | \( 10890 = 2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 446490.gb
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 1230.h2, its twist by $33$.
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.