# Properties

 Label 113a Number of curves 2 Conductor 113 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("113.a1")

sage: E.isogeny_class()

## Elliptic curves in class 113a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
113.a2 113a1 [1, 1, 1, 3, -4]  6 $$\Gamma_0(N)$$-optimal
113.a1 113a2 [1, 1, 1, -2, -2]  12

## Rank

sage: E.rank()

The elliptic curves in class 113a have rank $$0$$.

## Modular form113.2.a.a

sage: E.q_eigenform(10)

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

## 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 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. 