Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-485184x+138387184\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-485184xz^2+138387184z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-485184x+138387184\)
|
(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 51984 \) | = | $2^{4} \cdot 3^{2} \cdot 19^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-963540302564929536$ | = | $-1 \cdot 2^{12} \cdot 3^{6} \cdot 19^{9} $ |
|
| j-invariant: | $j$ | = | \( -\frac{89915392}{6859} \) | = | $-1 \cdot 2^{18} \cdot 7^{3} \cdot 19^{-3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1988058271344833429418407210$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.51586698734273704217752773487$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0331037033479094$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.698013445391969$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.27332337067132227946412509312$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.1865869653705782357130007450 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.186586965 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.273323 \cdot 1.000000 \cdot 8}{1^2} \\ & \approx 2.186586965\end{aligned}$$
Modular invariants
Modular form 51984.2.a.i
For more coefficients, see the Downloads section to the right.
| Modular degree: | 622080 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $19$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 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 | $\ell$-adic index |
|---|---|---|---|
| $3$ | 3Cs | 9.36.0.2 | $36$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 1999 & 54 \\ 1998 & 55 \end{array}\right),\left(\begin{array}{rr} 1025 & 0 \\ 0 & 2051 \end{array}\right),\left(\begin{array}{rr} 2009 & 2022 \\ 1212 & 1967 \end{array}\right),\left(\begin{array}{rr} 892 & 27 \\ 15 & 676 \end{array}\right),\left(\begin{array}{rr} 935 & 790 \\ 714 & 553 \end{array}\right),\left(\begin{array}{rr} 2033 & 1998 \\ 1260 & 1583 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 27 \\ 27 & 730 \end{array}\right),\left(\begin{array}{rr} 43 & 30 \\ 1866 & 1111 \end{array}\right),\left(\begin{array}{rr} 19 & 54 \\ 792 & 1495 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 1079 & 1998 \\ 1080 & 1997 \end{array}\right)$.
The torsion field $K:=\Q(E[2052])$ is a degree-$2872143360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2052\Z)$.
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 | $2$ | \( 3249 = 3^{2} \cdot 19^{2} \) |
| $3$ | additive | $2$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 144 = 2^{4} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 51984cw
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 19a1, its twist by $-228$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{19}) \) | \(\Z/3\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-57}) \) | \(\Z/3\Z\) | 2.0.228.1-19.1-d2 |
| $3$ | 3.1.76.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{19})\) | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.109744.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.1755904.2 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.47409408.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.2247651966910464.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.3083198857216.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.2247651966910464.3 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.10115714392459962143853624744311783424.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.78331280969964101283143221248.1 | \(\Z/9\Z\) | 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 | add | add | ord | ord | ord | ord | ord | add | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 0 | 2 | 0 | 0 | 0 | - | 0,0 | 0 | 0 | 0 | 0 | 2 | 0 |
| $\mu$-invariant(s) | - | - | 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 $0$.