Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2-27917x-2096220\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z-27917xz^2-2096220z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-2261304x-1521360495\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(196, 0)$ | $0$ | $2$ |
Integral points
\( \left(196, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 47652 \) | = | $2^{2} \cdot 3 \cdot 11 \cdot 19^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-488930151981744$ | = | $-1 \cdot 2^{4} \cdot 3^{10} \cdot 11 \cdot 19^{6} $ |
|
| j-invariant: | $j$ | = | \( -\frac{3196715008}{649539} \) | = | $-1 \cdot 2^{17} \cdot 3^{-10} \cdot 11^{-1} \cdot 29^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5410087923844131786093719279$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.16225975738545548786755249520$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0879123577519636$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.957185899763592$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.18267534631808678488751902940$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 60 $ = $ 3\cdot( 2 \cdot 5 )\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.7401301947713017733127854410 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.740130195 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.182675 \cdot 1.000000 \cdot 60}{2^2} \\ & \approx 2.740130195\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 213840 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $19$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 132 = 2^{2} \cdot 3 \cdot 11 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 89 & 4 \\ 46 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 37 & 100 \\ 32 & 99 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 26 & 1 \\ 119 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 129 & 4 \\ 128 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[132])$ is a degree-$5068800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/132\Z)$.
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 | $2$ | \( 3971 = 11 \cdot 19^{2} \) |
| $3$ | split multiplicative | $4$ | \( 15884 = 2^{2} \cdot 11 \cdot 19^{2} \) |
| $5$ | good | $2$ | \( 15884 = 2^{2} \cdot 11 \cdot 19^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 4332 = 2^{2} \cdot 3 \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 132 = 2^{2} \cdot 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 47652.j
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 132.a2, its twist by $-19$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-11}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.2287296.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.8510850702249984.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.633038481985536.82 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 11 | 19 |
|---|---|---|---|---|---|
| Reduction type | add | split | ord | nonsplit | add |
| $\lambda$-invariant(s) | - | 9 | 2 | 0 | - |
| $\mu$-invariant(s) | - | 0 | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ 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 $0$.