Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-x^2+2365x-7451\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-x^2z+2365xz^2-7451z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+37845x-439002\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1251413/71824, 3615521393/19248832)$ | $11.861884676394009705223969760$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 136242 \) | = | $2 \cdot 3^{4} \cdot 29^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-867252402018$ | = | $-1 \cdot 2 \cdot 3^{6} \cdot 29^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{3375}{2} \) | = | $2^{-1} \cdot 3^{3} \cdot 5^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.98006091934322054286979989518$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.2528931399840713164194587395$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.4265653296335434$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.9537340574685755$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $11.861884676394009705223969760$ |
|
| Real period: | $\Omega$ | ≈ | $0.52007280052836789244344236874$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\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
$$\begin{aligned} 12.338087166 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.520073 \cdot 11.861885 \cdot 2}{1^2} \\ & \approx 12.338087166\end{aligned}$$
Modular invariants
Modular form 136242.2.a.bj
For more coefficients, see the Downloads section to the right.
| Modular degree: | 145152 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $1$ | $IV$ | 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 |
|---|---|---|
| $2$ | 2G | 8.2.0.1 |
| $3$ | 3B | 3.4.0.1 |
| $7$ | 7B | 7.8.0.1 |
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
$\left(\begin{array}{rr} 10585 & 4032 \\ 10584 & 10585 \end{array}\right),\left(\begin{array}{rr} 2553 & 7714 \\ 6496 & 10209 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2436 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4032 & 1 \end{array}\right),\left(\begin{array}{rr} 3046 & 6873 \\ 12789 & 9745 \end{array}\right),\left(\begin{array}{rr} 13399 & 6090 \\ 4263 & 1219 \end{array}\right),\left(\begin{array}{rr} 1 & 9744 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6264 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10583 & 0 \\ 0 & 14615 \end{array}\right),\left(\begin{array}{rr} 1 & 6090 \\ 0 & 2089 \end{array}\right),\left(\begin{array}{rr} 4873 & 9744 \\ 4872 & 9745 \end{array}\right),\left(\begin{array}{rr} 8527 & 3306 \\ 2436 & 13399 \end{array}\right),\left(\begin{array}{rr} 11369 & 12992 \\ 1624 & 12993 \end{array}\right),\left(\begin{array}{rr} 12181 & 6090 \\ 12789 & 6091 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8526 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 7482 \\ 6090 & 7309 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 9744 & 1 \end{array}\right)$.
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 | $6$ | \( 58 = 2 \cdot 29 \) |
| $29$ | 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 162c1, 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 |
| $6$ | 6.0.480048687.1 | \(\Z/7\Z\) | 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 |
| $12$ | 12.0.230446741890423969.1 | \(\Z/21\Z\) | 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 |
| $18$ | 18.0.28999851931841573471329636319232.1 | \(\Z/14\Z\) | not in database |
| $18$ | 18.0.36702937601236991424651570966528.1 | \(\Z/21\Z\) | 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.