# Properties

 Label 2112.y Number of curves $4$ Conductor $2112$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("2112.y1")

sage: E.isogeny_class()

## Elliptic curves in class 2112.y

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2112.y1 2112y3 [0, 1, 0, -2817, -58497] [2] 1536
2112.y2 2112y4 [0, 1, 0, -417, 1887] [2] 1536
2112.y3 2112y2 [0, 1, 0, -177, -945] [2, 2] 768
2112.y4 2112y1 [0, 1, 0, 3, -45] [2] 384 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form2112.2.a.y

sage: E.q_eigenform(10)

$$q + q^{3} + 2q^{5} - 4q^{7} + q^{9} - q^{11} - 6q^{13} + 2q^{15} + 6q^{17} - 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.