# Properties

 Label 112a Number of curves 2 Conductor 112 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("112.a1")
sage: E.isogeny_class()

## Elliptic curves in class 112a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
112.a2 112a1 [0, 1, 0, 0, 4] 2 8 $$\Gamma_0(N)$$-optimal
112.a1 112a2 [0, 1, 0, -40, 84] 2 16

## Rank

sage: E.rank()

The elliptic curves in class 112a have rank $$1$$.

## Modular form112.2.a.a

sage: E.q_eigenform(10)
$$q - 2q^{3} - 4q^{5} - q^{7} + q^{9} + 8q^{15} - 2q^{17} + 2q^{19} + O(q^{20})$$

## 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. 