Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-1665947x+820144550\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-1665947xz^2+820144550z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-2159066691x+38271141336510\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(534, 8836)$ | $0.77613725710462732747492797893$ | $\infty$ |
| $(-1489, 744)$ | $0$ | $2$ |
| $(687, -344)$ | $0$ | $2$ |
Integral points
\( \left(-1489, 744\right) \), \( \left(-180, 33469\right) \), \( \left(-180, -33290\right) \), \( \left(534, 8836\right) \), \( \left(534, -9371\right) \), \( \left(687, -344\right) \), \( \left(912, 7261\right) \), \( \left(912, -8174\right) \), \( \left(975, 10600\right) \), \( \left(975, -11576\right) \), \( \left(2319, 95944\right) \), \( \left(2319, -98264\right) \), \( \left(3186, 165304\right) \), \( \left(3186, -168491\right) \), \( \left(87215, 25710120\right) \), \( \left(87215, -25797336\right) \)
Invariants
| Conductor: | $N$ | = | \( 133518 \) | = | $2 \cdot 3 \cdot 7 \cdot 11 \cdot 17^{2}$ |
|
| Discriminant: | $\Delta$ | = | $5234781315311944704$ | = | $2^{10} \cdot 3^{6} \cdot 7^{4} \cdot 11^{2} \cdot 17^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{21184262604460873}{216872764416} \) | = | $2^{-10} \cdot 3^{-6} \cdot 7^{-4} \cdot 11^{-2} \cdot 19^{3} \cdot 14563^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4093382368493559183099953864$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.99273156482124787818522807746$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.003703237419837$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.6256019307388625$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.77613725710462732747492797893$ |
|
| Real period: | $\Omega$ | ≈ | $0.24298674774457081268168037599$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 384 $ = $ 2\cdot( 2 \cdot 3 )\cdot2^{2}\cdot2\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.5261856297738843494086195779 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.526185630 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.242987 \cdot 0.776137 \cdot 384}{4^2} \\ & \approx 4.526185630\end{aligned}$$
Modular invariants
Modular form 133518.2.a.z
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4423680 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $3$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $7$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $17$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4488 = 2^{3} \cdot 3 \cdot 11 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 2993 & 2380 \\ 442 & 273 \end{array}\right),\left(\begin{array}{rr} 2245 & 3434 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3367 & 3434 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 171 & 3434 \\ 238 & 1055 \end{array}\right),\left(\begin{array}{rr} 3431 & 0 \\ 0 & 4487 \end{array}\right),\left(\begin{array}{rr} 4485 & 4 \\ 4484 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[4488])$ is a degree-$1588278067200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4488\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 289 = 17^{2} \) |
| $3$ | split multiplicative | $4$ | \( 44506 = 2 \cdot 7 \cdot 11 \cdot 17^{2} \) |
| $5$ | good | $2$ | \( 66759 = 3 \cdot 7 \cdot 11 \cdot 17^{2} \) |
| $7$ | split multiplicative | $8$ | \( 19074 = 2 \cdot 3 \cdot 11 \cdot 17^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 12138 = 2 \cdot 3 \cdot 7 \cdot 17^{2} \) |
| $17$ | additive | $146$ | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 133518.z
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 462.c2, its twist by $17$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{22}, \sqrt{-34})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{34})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{33}, \sqrt{51})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | split | ord | split | nonsplit | ord | add | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 10 | 4 | 3 | 4 | 1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.