# Properties

 Label 2112a Number of curves 2 Conductor 2112 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("2112.h1")

sage: E.isogeny_class()

## Elliptic curves in class 2112a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2112.h1 2112a1 [0, -1, 0, -33, 33]  256 $$\Gamma_0(N)$$-optimal
2112.h2 2112a2 [0, -1, 0, 127, 129]  512

## Rank

sage: E.rank()

The elliptic curves in class 2112a have rank $$1$$.

## Modular form2112.2.a.h

sage: E.q_eigenform(10)

$$q - q^{3} + 2q^{7} + q^{9} - q^{11} - 2q^{17} - 8q^{19} + O(q^{20})$$

## 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. 