Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-592824x+175883580\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-592824xz^2+175883580z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-48018771x+128075073534\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(426, 684)$ | $0$ | $4$ |
Integral points
\((426,\pm 684)\), \( \left(445, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 3192 \) | = | $2^{3} \cdot 3 \cdot 7 \cdot 19$ |
|
| Discriminant: | $\Delta$ | = | $3707605343232$ | = | $2^{10} \cdot 3^{4} \cdot 7^{3} \cdot 19^{4} $ |
|
| j-invariant: | $j$ | = | \( \frac{22501000029889239268}{3620708343} \) | = | $2^{2} \cdot 3^{-4} \cdot 7^{-3} \cdot 19^{-4} \cdot 1778473^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8141540726524812049530872321$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2365314221858601137720604642$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0120375317308423$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.381877987241211$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.61750109449594611849769173845$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot1\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L(E,1)$ | ≈ | $0.61750109449594611849769173845 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 0.617501094 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.617501 \cdot 1.000000 \cdot 16}{4^2} \\ & \approx 0.617501094\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 24576 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | 1 | 3 | 10 | 0 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $7$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $19$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
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 | 4.12.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1064 = 2^{3} \cdot 7 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 920 & 3 \\ 309 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 1057 & 8 \\ 1056 & 9 \end{array}\right),\left(\begin{array}{rr} 1009 & 8 \\ 844 & 33 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1058 & 1059 \end{array}\right),\left(\begin{array}{rr} 403 & 402 \\ 146 & 675 \end{array}\right),\left(\begin{array}{rr} 400 & 939 \\ 403 & 432 \end{array}\right)$.
The torsion field $K:=\Q(E[1064])$ is a degree-$7942717440$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1064\Z)$.
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 | $2$ | \( 7 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 152 = 2^{3} \cdot 19 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 456 = 2^{3} \cdot 3 \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 168 = 2^{3} \cdot 3 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 3192c
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.161728.1 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.30118144.2 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.1004806820921344.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.20506261651456.8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.23640037567488.17 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 7 | 19 |
|---|---|---|---|---|
| Reduction type | add | nonsplit | nonsplit | split |
| $\lambda$-invariant(s) | - | 0 | 0 | 1 |
| $\mu$-invariant(s) | - | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ 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 $0$.