\(y^2+xy=x^3+x^2-2x\) y^2+xy=x^3+x^2-2x	(homogenize, simplify)
\(y^2z+xyz=x^3+x^2z-2xz^2\) y^2z+xyz=x^3+x^2z-2xz^2	(dehomogenize, simplify)
\(y^2=x^3-3267x+45630\) y^2=x^3-3267x+45630	(homogenize, minimize)

Mordell-Weil group structure

\(\Z \oplus \Z/{2}\Z\)

Infinite order Mordell-Weil generator and height

$P$	=	\(\left(2, 2\right)\) (2, 2)
$\hat{h}(P)$	≈	$0.14325389294088007147627441303$

Torsion generators

\( \left(0, 0\right) \) (0, 0)

Integral points

Invariants

BSD invariants

BSD formula

$\displaystyle 0.677284898 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 4.727864 \cdot 0.143254 \cdot 4}{2^2} \approx 0.677284898$

Modular invariants

Modular form   102.2.a.a

\( q - q^{2} - q^{3} + q^{4} - 4 q^{5} + q^{6} - 2 q^{7} - q^{8} + q^{9} + 4 q^{10} - q^{12} - 6 q^{13} + 2 q^{14} + 4 q^{15} + q^{16} - q^{17} - q^{18} + 4 q^{19} + O(q^{20}) \) q - q^2 - q^3 + q^4 - 4*q^5 + q^6 - 2*q^7 - q^8 + q^9 + 4*q^10 - q^12 - 6*q^13 + 2*q^14 + 4*q^15 + q^16 - q^17 - q^18 + 4*q^19 + O(q^20)

For more coefficients, see the Downloads section to the right.

Local data

This elliptic curve is semistable. There are 3 primes $p$ of bad reduction:

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 136 = 2^{3} \cdot 17 \), index $12$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 89 & 52 \\ 16 & 119 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 133 & 4 \\ 132 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 74 & 1 \\ 31 & 0 \end{array}\right),\left(\begin{array}{rr} 69 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.

The torsion field $K:=\Q(E[136])$ is a degree-$10027008$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/136\Z)$.

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 102.a consists of 2 curves linked by isogenies of degree 2.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

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

$p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

Additional information

This is the elliptic curve $E$ associated to the [Somos-5 sequence] $\{a(n)\}$. Let $T$ be the $2$-torsion point $(0,0)$, and $P$ the point $(2,2)$ such that $E(\Q) = \Z P \oplus \{0, T\}$. Then the $x$- and $y$-coordinates of $nP+T$ have denominators $d_n^2$ and $d_n^3$ where $$d_n = 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217$$ for $1 \leq n \leq 10$, and $d_n = a(n+2)$ in general, satisfying the Somos-5 recurrence $$ d_n d_{n+5} = d_{n+1} d_{n+4} + d_{n+2} d_{n+3}. $$ Thus the regulator of $E$, which is the canonical height $\hat h(P) = 0.143\ldots$, controls the growth of the $a(n)$: asymptotically $\log a_n \sim \frac12 \hat h(P) n^2$.