# Properties

 Label 82.a Number of curves 2 Conductor 82 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("82.a1")

sage: E.isogeny_class()

## Elliptic curves in class 82.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
82.a1 82a2 [1, 0, 1, -12, -16]  8
82.a2 82a1 [1, 0, 1, -2, 0]  4 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 82.a have rank $$1$$.

## Modular form82.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - 2q^{3} + q^{4} - 2q^{5} + 2q^{6} - 4q^{7} - q^{8} + q^{9} + 2q^{10} - 2q^{11} - 2q^{12} + 4q^{13} + 4q^{14} + 4q^{15} + q^{16} - 2q^{17} - q^{18} + 6q^{19} + O(q^{20})$$

## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the LMFDB numbering.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels. 