# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("2112.n1")

sage: E.isogeny_class()

## Elliptic curves in class 2112.n

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2112.n1 2112d3 [0, -1, 0, -644161, -198779327]  19200
2112.n2 2112d4 [0, -1, 0, -643521, -199194687]  38400
2112.n3 2112d1 [0, -1, 0, -2881, 44353]  3840 $$\Gamma_0(N)$$-optimal
2112.n4 2112d2 [0, -1, 0, 7359, 279873]  7680

## Rank

sage: E.rank()

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

## Modular form2112.2.a.n

sage: E.q_eigenform(10)

$$q - q^{3} + 4q^{5} - 2q^{7} + q^{9} - q^{11} - 4q^{13} - 4q^{15} - 2q^{17} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 