# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("2112.d1")

sage: E.isogeny_class()

## Elliptic curves in class 2112w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2112.d3 2112w1 [0, -1, 0, -49, 145]  256 $$\Gamma_0(N)$$-optimal
2112.d2 2112w2 [0, -1, 0, -129, -351] [2, 2] 512
2112.d1 2112w3 [0, -1, 0, -1889, -30975]  1024
2112.d4 2112w4 [0, -1, 0, 351, -2751]  1024

## Rank

sage: E.rank()

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

## Modular form2112.2.a.d

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. 