Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-x^2-3861059x+2907266627\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-x^2z-3861059xz^2+2907266627z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-61776939x+186003287206\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{9355}{9}, \frac{94502}{27}\right) \) | $7.1900357379265081853595745312$ | $\infty$ |
| \( \left(1387, 14098\right) \) | $0$ | $3$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([28065:94502:27]\) | $7.1900357379265081853595745312$ | $\infty$ |
| \([1387:14098:1]\) | $0$ | $3$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{37411}{9}, \frac{868384}{27}\right) \) | $7.1900357379265081853595745312$ | $\infty$ |
| \( \left(5547, 118336\right) \) | $0$ | $3$ |
Integral points
\( \left(1387, 14098\right) \), \( \left(1387, -15486\right) \)
\([1387:14098:1]\), \([1387:-15486:1]\)
\((5547,\pm 118336)\)
Invariants
| Conductor: | $N$ | = | \( 33282 \) | = | $2 \cdot 3^{2} \cdot 43^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $34900779017712144384$ | = | $2^{12} \cdot 3^{6} \cdot 43^{8} $ |
|
| j-invariant: | $j$ | = | \( \frac{747081097}{4096} \) | = | $2^{-12} \cdot 7^{3} \cdot 37^{3} \cdot 43$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5931360639067083262417876572$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.46363682422305480177106330349$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9582950471368006$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.484894801992146$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.1900357379265081853595745312$ |
|
| Real period: | $\Omega$ | ≈ | $0.20764635241911063661887528258$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 36 $ = $ ( 2^{2} \cdot 3 )\cdot1\cdot3 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $5.9719387789739516968915961022 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 5.971938779 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.207646 \cdot 7.190036 \cdot 36}{3^2} \\ & \approx 5.971938779\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1300320 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $3$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $43$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 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 |
|---|---|---|---|
| $2$ | 2Cn | 2.2.0.1 | $2$ |
| $3$ | 3B.1.1 | 3.8.0.1 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 516 = 2^{2} \cdot 3 \cdot 43 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 480 & 511 \\ 425 & 47 \end{array}\right),\left(\begin{array}{rr} 343 & 504 \\ 344 & 515 \end{array}\right),\left(\begin{array}{rr} 259 & 12 \\ 6 & 73 \end{array}\right),\left(\begin{array}{rr} 7 & 12 \\ 450 & 403 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 478 & 501 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 505 & 12 \\ 504 & 13 \end{array}\right)$.
The torsion field $K:=\Q(E[516])$ is a degree-$160199424$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/516\Z)$.
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 | $4$ | \( 16641 = 3^{2} \cdot 43^{2} \) |
| $3$ | additive | $6$ | \( 1849 = 43^{2} \) |
| $43$ | additive | $674$ | \( 18 = 2 \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 33282y
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 3698b2, its twist by $129$.
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 |
|---|---|---|---|
| $3$ | 3.3.1849.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.7476917787.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.3.645010848023732928.4 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.417991852000843034688232562403.1 | \(\Z/6\Z \oplus \Z/6\Z\) | 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 | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | ord | add | ss |
| $\lambda$-invariant(s) | 2 | - | 1 | 1 | 3,1 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | - | 1,1 |
| $\mu$-invariant(s) | 0 | - | 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.