Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2-3435318x+2451690342\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z-3435318xz^2+2451690342z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-54965091x+156853216798\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1027, 1943)$ | $0$ | $3$ |
Integral points
\( \left(1027, 1943\right) \), \( \left(1027, -2970\right) \)
Invariants
| Conductor: | $N$ | = | \( 2142 \) | = | $2 \cdot 3^{2} \cdot 7 \cdot 17$ |
|
| Discriminant: | $\Delta$ | = | $-177915256526672154$ | = | $-1 \cdot 2 \cdot 3^{7} \cdot 7^{3} \cdot 17^{9} $ |
|
| j-invariant: | $j$ | = | \( -\frac{6150311179917589675873}{244053849830826} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-1} \cdot 7^{-3} \cdot 17^{-9} \cdot 37^{3} \cdot 495181^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3934166814767256485980895203$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.8441105371426708029004669018$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0370184575562917$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.401077916266199$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.30078619337178501419768532738$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 54 $ = $ 1\cdot2\cdot3\cdot3^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.8047171602307100851861119643 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 1.804717160 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.300786 \cdot 1.000000 \cdot 54}{3^2} \\ & \approx 1.804717160\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 77760 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $17$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
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 |
|---|---|---|
| $3$ | 3B.1.1 | 9.72.0.6 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8568 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 17 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4285 & 18 \\ 4293 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 6121 & 18 \\ 3681 & 163 \end{array}\right),\left(\begin{array}{rr} 5545 & 18 \\ 7065 & 163 \end{array}\right),\left(\begin{array}{rr} 8551 & 18 \\ 8550 & 19 \end{array}\right),\left(\begin{array}{rr} 2141 & 4266 \\ 2133 & 2455 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 2143 & 18 \\ 2151 & 163 \end{array}\right)$.
The torsion field $K:=\Q(E[8568])$ is a degree-$6549481193472$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8568\Z)$.
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 | $4$ | \( 1071 = 3^{2} \cdot 7 \cdot 17 \) |
| $3$ | additive | $8$ | \( 2 \) |
| $7$ | split multiplicative | $8$ | \( 306 = 2 \cdot 3^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 2142j
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 714i3, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.2856.1 | \(\Z/6\Z\) | not in database |
| $3$ | \(\Q(\zeta_{9})^+\) | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.23295638016.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.2834352.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.2834352.4 | \(\Z/9\Z\) | not in database |
| $9$ | 9.3.1375584129206784.2 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.16599265906765726789632.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.17582592502916589732558131885417583280128.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.1059407544397173007102900264639660032.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.1059407544397173007102900264639660032.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.4897860516073513491019249127736364498944.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | add | ord | split | ord | ord | split | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | - | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-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
All $p$-adic regulators are identically $1$ since the rank is $0$.