# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("2112.e1")

sage: E.isogeny_class()

## Elliptic curves in class 2112.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2112.e1 2112u3 [0, -1, 0, -22529, 1309089]  3072
2112.e2 2112u2 [0, -1, 0, -1409, 20769] [2, 2] 1536
2112.e3 2112u4 [0, -1, 0, -769, 39073]  3072
2112.e4 2112u1 [0, -1, 0, -129, 33]  768 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form2112.2.a.e

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 