# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("2112.p1")

sage: E.isogeny_class()

## Elliptic curves in class 2112z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2112.p1 2112z1 [0, 1, 0, -32065, 2199359]  5376 $$\Gamma_0(N)$$-optimal
2112.p2 2112z2 [0, 1, 0, -31905, 2222559]  10752

## Rank

sage: E.rank()

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

## Modular form2112.2.a.p

sage: E.q_eigenform(10)

$$q + q^{3} - 4q^{5} + 2q^{7} + q^{9} - q^{11} - 4q^{15} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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. 