# Properties

 Label 2112m Number of curves 4 Conductor 2112 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("2112.s1")

sage: E.isogeny_class()

## Elliptic curves in class 2112m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2112.s3 2112m1 [0, 1, 0, -49, -145]  256 $$\Gamma_0(N)$$-optimal
2112.s2 2112m2 [0, 1, 0, -129, 351] [2, 2] 512
2112.s1 2112m3 [0, 1, 0, -1889, 30975]  1024
2112.s4 2112m4 [0, 1, 0, 351, 2751]  1024

## Rank

sage: E.rank()

The elliptic curves in class 2112m have rank $$0$$.

## Modular form2112.2.a.s

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{5} + q^{9} - q^{11} - 2q^{13} - 2q^{15} + 6q^{17} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 