Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2-22230207x-40252036811\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z-22230207xz^2-40252036811z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-355683315x-2576486039218\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-10477/4, 10477/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 33462 \) | = | $2 \cdot 3^{2} \cdot 11 \cdot 13^{2}$ |
|
| Discriminant: | $\Delta$ | = | $2958200193782178022392$ | = | $2^{3} \cdot 3^{9} \cdot 11^{6} \cdot 13^{9} $ |
|
| j-invariant: | $j$ | = | \( \frac{157158018407125}{382657176} \) | = | $2^{-3} \cdot 3^{-3} \cdot 5^{3} \cdot 11^{-6} \cdot 43^{3} \cdot 251^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9984204383755049766595378053$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.52540227594529757892179960567$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9993281682598464$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.9861296483187925$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.069538804769777644404750098356$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 1\cdot2^{2}\cdot( 2 \cdot 3 )\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $0.83446565723733173285700118027 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 0.834465657 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.069539 \cdot 1.000000 \cdot 48}{2^2} \\ & \approx 0.834465657\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1797120 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $11$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $13$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3Ns | 3.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \), index $144$, genus $5$, and generators
$\left(\begin{array}{rr} 7 & 12 \\ 144 & 247 \end{array}\right),\left(\begin{array}{rr} 804 & 5 \\ 2995 & 3360 \end{array}\right),\left(\begin{array}{rr} 5 & 12 \\ 3384 & 3317 \end{array}\right),\left(\begin{array}{rr} 6 & 577 \\ 1759 & 36 \end{array}\right),\left(\begin{array}{rr} 937 & 12 \\ 2190 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 3421 & 12 \\ 3420 & 13 \end{array}\right),\left(\begin{array}{rr} 10 & 3 \\ 1689 & 3424 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1574 & 3015 \\ 3003 & 3290 \end{array}\right)$.
The torsion field $K:=\Q(E[3432])$ is a degree-$177124147200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3432\Z)$.
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 | $4$ | \( 117 = 3^{2} \cdot 13 \) |
| $3$ | additive | $2$ | \( 169 = 13^{2} \) |
| $11$ | split multiplicative | $12$ | \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $62$ | \( 198 = 2 \cdot 3^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 33462bn
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 11154l2, its twist by $-39$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{78}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.25520352.1 | \(\Z/4\Z\) | not in database |
| $4$ | 4.2.6591.1 | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.19773.1 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.390971529.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.4.1601419382784.23 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.1601419382784.39 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | 11 | 13 |
|---|---|---|---|---|
| Reduction type | nonsplit | add | split | add |
| $\lambda$-invariant(s) | 3 | - | 1 | - |
| $\mu$-invariant(s) | 0 | - | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.