# Properties

 Label 102a Number of curves 2 Conductor 102 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("102.a1")

sage: E.isogeny_class()

## Elliptic curves in class 102a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
102.a1 102a1 [1, 1, 0, -2, 0]  8 $$\Gamma_0(N)$$-optimal
102.a2 102a2 [1, 1, 0, 8, 10]  16

## Rank

sage: E.rank()

The elliptic curves in class 102a have rank $$1$$.

## Modular form102.2.a.a

sage: E.q_eigenform(10)

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

## 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 Cremona 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 Cremona labels. 