# Properties

 Label 2112.v Number of curves 4 Conductor 2112 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("2112.v1")

sage: E.isogeny_class()

## Elliptic curves in class 2112.v

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2112.v1 2112x3 [0, 1, 0, -5153, 140127]  2304
2112.v2 2112x4 [0, 1, 0, -2593, 281951]  4608
2112.v3 2112x1 [0, 1, 0, -353, -2529]  768 $$\Gamma_0(N)$$-optimal
2112.v4 2112x2 [0, 1, 0, 287, -10081]  1536

## Rank

sage: E.rank()

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

## Modular form2112.2.a.v

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{7} + q^{9} - q^{11} + 4q^{13} - 6q^{17} - 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels. 