Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-95x-697\)
|
(homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-95xz^2-697z^3\)
|
(dehomogenize, simplify) |
\(y^2=x^3-1515x-46106\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$(19, 54)$ | $0$ | $3$ |
Integral points
\( \left(19, 54\right) \), \( \left(19, -74\right) \)
Invariants
Conductor: | $N$ | = | \( 162 \) | = | $2 \cdot 3^{4}$ |
|
Discriminant: | $\Delta$ | = | $-169869312$ | = | $-1 \cdot 2^{21} \cdot 3^{4} $ |
|
j-invariant: | $j$ | = | \( -\frac{1159088625}{2097152} \) | = | $-1 \cdot 2^{-21} \cdot 3^{2} \cdot 5^{3} \cdot 101^{3}$ |
|
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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.26936807887764018183084025072$ |
|
||
Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.096836017345063048634241494921$ |
|
||
$abc$ quality: | $Q$ | ≈ | $1.1123490200903752$ | |||
Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.244672406658039$ |
BSD invariants
Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
Mordell-Weil rank: | $r$ | = | $ 0$ |
|
Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
Real period: | $\Omega$ | ≈ | $0.72170105890730159936550912938$ |
|
Tamagawa product: | $\prod_{p}c_p$ | = | $ 21 $ = $ ( 3 \cdot 7 )\cdot1 $ |
|
Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
|
Special value: | $ L(E,1)$ | ≈ | $1.6839691374503703985195213019 $ |
|
Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 1.683969137 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.721701 \cdot 1.000000 \cdot 21}{3^2} \\ & \approx 1.683969137\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 42 |
|
$ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $21$ | $I_{21}$ | split multiplicative | -1 | 1 | 21 | 21 |
$3$ | $1$ | $II$ | additive | 1 | 4 | 4 | 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 |
---|---|---|
$2$ | 2G | 8.2.0.1 |
$3$ | 3B.1.1 | 3.8.0.1 |
$7$ | 7B.2.3 | 7.16.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \), index $768$, genus $21$, and generators
$\left(\begin{array}{rr} 281 & 392 \\ 112 & 393 \end{array}\right),\left(\begin{array}{rr} 73 & 402 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 426 \\ 42 & 253 \end{array}\right),\left(\begin{array}{rr} 1 & 216 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 462 & 1 \end{array}\right),\left(\begin{array}{rr} 85 & 42 \\ 189 & 43 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 168 & 1 \end{array}\right),\left(\begin{array}{rr} 337 & 168 \\ 336 & 169 \end{array}\right),\left(\begin{array}{rr} 295 & 42 \\ 231 & 211 \end{array}\right),\left(\begin{array}{rr} 22 & 321 \\ 189 & 169 \end{array}\right),\left(\begin{array}{rr} 249 & 226 \\ 448 & 345 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 420 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 168 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 463 & 282 \\ 420 & 295 \end{array}\right)$.
The torsion field $K:=\Q(E[504])$ is a degree-$15676416$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/504\Z)$.
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 | $4$ | \( 81 = 3^{4} \) |
$3$ | additive | $8$ | \( 1 \) |
$7$ | good | $2$ | \( 81 = 3^{4} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 7 and 21.
Its isogeny class 162.c
consists of 4 curves linked by isogenies of
degrees dividing 21.
Twists
The minimal quadratic twist of this elliptic curve is 162.b2, its twist by $-3$.
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 |
---|---|---|---|
$3$ | 3.1.648.1 | \(\Z/6\Z\) | not in database |
$6$ | 6.0.3359232.4 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$6$ | 6.0.177147.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
$6$ | 6.0.110270727.2 | \(\Z/21\Z\) | not in database |
$9$ | 9.3.74384733888.1 | \(\Z/9\Z\) | not in database |
$12$ | 12.2.5777633090469888.10 | \(\Z/12\Z\) | not in database |
$18$ | 18.0.1062353018033006514536448.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
$18$ | 18.0.351496200956998572502045949952.2 | \(\Z/42\Z\) | not in database |
$21$ | 21.3.12777737809210143774260519108727.3 | \(\Z/21\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 |
---|---|---|---|---|
Reduction type | split | add | ss | ord |
$\lambda$-invariant(s) | 4 | - | 0,0 | 0 |
$\mu$-invariant(s) | 0 | - | 0,0 | 1 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ 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 $0$.