Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2+18x-11\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z+18xz^2-11z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+1431x-12339\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(9, 31)$ | $0.17351023129422475915007959768$ | $\infty$ |
Integral points
\((1,\pm 3)\), \((9,\pm 31)\), \((753,\pm 20677)\)
Invariants
| Conductor: | $N$ | = | \( 124 \) | = | $2^{2} \cdot 31$ |
|
| Discriminant: | $\Delta$ | = | $-476656$ | = | $-1 \cdot 2^{4} \cdot 31^{3} $ |
|
| j-invariant: | $j$ | = | \( \frac{38112512}{29791} \) | = | $2^{8} \cdot 31^{-3} \cdot 53^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.22245546009287257183602063064$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.45350452027952100830843133779$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.887542930342586$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.196568526560052$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.17351023129422475915007959768$ |
|
| Real period: | $\Omega$ | ≈ | $1.6444424111396318194918200617$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 3 $ = $ 1\cdot3 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $0.85598274932061048782613103328 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 0.855982749 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.644442 \cdot 0.173510 \cdot 3}{1^2} \\ & \approx 0.855982749\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $31$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 186 = 2 \cdot 3 \cdot 31 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 158 & 33 \\ 103 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 181 & 6 \\ 180 & 7 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 127 & 6 \\ 9 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[186])$ is a degree-$16070400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/186\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 31 \) |
| $3$ | good | $2$ | \( 4 = 2^{2} \) |
| $31$ | split multiplicative | $32$ | \( 4 = 2^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 124a
consists of 2 curves linked by isogenies of
degree 3.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | 2.0.3.1-15376.2-a2 |
| $3$ | 3.1.31.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.108.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.29791.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.34992.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.25947.1 | \(\Z/6\Z\) | not in database |
| $9$ | 9.1.37528080192.1 | \(\Z/6\Z\) | not in database |
| $12$ | 12.2.7212266713513984.11 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.646990183449.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.24602211709238575976815410946486272.4 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.38025633678223934435328.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.41956357515109971509735424.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ord | ord | ord | ord | ord | ord | ord | ord | ss | split | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 2 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | - | 1 | 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.