Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-4957x+151618\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-4957xz^2+151618z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-401544x+109324917\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-82, 0)$ | $0$ | $2$ |
Integral points
\( \left(-82, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 14196 \) | = | $2^{2} \cdot 3 \cdot 7 \cdot 13^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-1918598655792$ | = | $-1 \cdot 2^{4} \cdot 3 \cdot 7^{2} \cdot 13^{8} $ |
|
| j-invariant: | $j$ | = | \( -\frac{174456832}{24843} \) | = | $-1 \cdot 2^{17} \cdot 3^{-1} \cdot 7^{-2} \cdot 11^{3} \cdot 13^{-2}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0878132542605554324202076011$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.42571048465686137207894682684$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9271646887473219$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.907601665577724$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.80449840033197885435178823994$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot1\cdot2\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.6089968006639577087035764799 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 1.608996801 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.804498 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 1.608996801\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 24192 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
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 \( 1092 = 2^{2} \cdot 3 \cdot 7 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 277 & 820 \\ 272 & 819 \end{array}\right),\left(\begin{array}{rr} 730 & 1 \\ 727 & 0 \end{array}\right),\left(\begin{array}{rr} 925 & 4 \\ 758 & 9 \end{array}\right),\left(\begin{array}{rr} 157 & 4 \\ 314 & 9 \end{array}\right),\left(\begin{array}{rr} 1089 & 4 \\ 1088 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[1092])$ is a degree-$20288765952$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1092\Z)$.
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 | $2$ | \( 507 = 3 \cdot 13^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 84 = 2^{2} \cdot 3 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 14196.g
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1092.a2, its twist by $13$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.1589952.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.4178851762176.27 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.22751526260736.4 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | 7 | 13 |
|---|---|---|---|---|
| Reduction type | add | nonsplit | nonsplit | add |
| $\lambda$-invariant(s) | - | 0 | 0 | - |
| $\mu$-invariant(s) | - | 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 $0$.