Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-x^2-905915x+332105131\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-x^2z-905915xz^2+332105131z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-14494635x+21240233766\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(689, 5542\right) \) | $1.6945549537705728150319956801$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([689:5542:1]\) | $1.6945549537705728150319956801$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(2755, 47096\right) \) | $1.6945549537705728150319956801$ | $\infty$ |
Integral points
\( \left(689, 5542\right) \), \( \left(689, -6232\right) \)
\([689:5542:1]\), \([689:-6232:1]\)
\((2755,\pm 47096)\)
Invariants
| Conductor: | $N$ | = | \( 136242 \) | = | $2 \cdot 3^{4} \cdot 29^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-55504153729152$ | = | $-1 \cdot 2^{7} \cdot 3^{6} \cdot 29^{6} $ |
|
| j-invariant: | $j$ | = | \( -\frac{189613868625}{128} \) | = | $-1 \cdot 2^{-7} \cdot 3^{3} \cdot 5^{3} \cdot 383^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9530159938708771954224762669$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.27993806545641466386678236774$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.1259568215438134$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.4631083070808435$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.6945549537705728150319956801$ |
|
| Real period: | $\Omega$ | ≈ | $0.52007280052836789244344236874$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 14 $ = $ 7\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $12.338087166393531074992943777 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 12.338087166 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.520073 \cdot 1.694555 \cdot 14}{1^2} \\ & \approx 12.338087166\end{aligned}$$
Modular invariants
Modular form 136242.2.a.bj
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1016064 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $3$ | $1$ | $IV$ | additive | -1 | 4 | 6 | 0 |
| $29$ | $2$ | $I_0^{*}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2G | 8.2.0.1 | $2$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
| $7$ | 7B | 7.8.0.1 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 14616 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 29 \), index $768$, genus $21$, and generators
$\left(\begin{array}{rr} 11369 & 12992 \\ 1624 & 12993 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4032 & 1 \end{array}\right),\left(\begin{array}{rr} 10585 & 4032 \\ 10584 & 10585 \end{array}\right),\left(\begin{array}{rr} 2089 & 1914 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13399 & 6090 \\ 4263 & 1219 \end{array}\right),\left(\begin{array}{rr} 12181 & 6090 \\ 12789 & 6091 \end{array}\right),\left(\begin{array}{rr} 1 & 9744 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2436 & 1 \end{array}\right),\left(\begin{array}{rr} 8527 & 3306 \\ 2436 & 13399 \end{array}\right),\left(\begin{array}{rr} 3046 & 6873 \\ 12789 & 9745 \end{array}\right),\left(\begin{array}{rr} 1 & 7482 \\ 6090 & 7309 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8526 & 1 \end{array}\right),\left(\begin{array}{rr} 10583 & 0 \\ 0 & 14615 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 9744 & 1 \end{array}\right),\left(\begin{array}{rr} 465 & 1450 \\ 6496 & 12297 \end{array}\right),\left(\begin{array}{rr} 1 & 6264 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4873 & 9744 \\ 4872 & 9745 \end{array}\right)$.
The torsion field $K:=\Q(E[14616])$ is a degree-$10692569825280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/14616\Z)$.
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 | $4$ | \( 68121 = 3^{4} \cdot 29^{2} \) |
| $3$ | additive | $6$ | \( 58 = 2 \cdot 29 \) |
| $7$ | good | $2$ | \( 68121 = 3^{4} \cdot 29^{2} \) |
| $29$ | additive | $422$ | \( 162 = 2 \cdot 3^{4} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 7 and 21.
Its isogeny class 136242e
consists of 4 curves linked by isogenies of
degrees dividing 21.
Twists
The minimal quadratic twist of this elliptic curve is 162c3, its twist by $29$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{29}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.648.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.3359232.4 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.853419888.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.6.8068178282409.1 | \(\Z/7\Z\) | not in database |
| $6$ | 6.2.10241038656.14 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/21\Z\) | not in database |
| $18$ | 18.6.406363455452895582493921990125470976.2 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.601340929658666867501491338715594752.1 | \(\Z/6\Z\) | not in database |
| $18$ | 18.6.137678580825657206070206700223618126690123776.1 | \(\Z/14\Z\) | 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 | split | add | ss | ord | ord | ord | ord | ord | ord | add | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | - | 3,1 | 1 | 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.