# Properties

 Label 42042.db Number of curves 2 Conductor 42042 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("42042.db1")
sage: E.isogeny_class()

## Elliptic curves in class 42042.db

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
42042.db1 42042dk2 [1, 0, 0, -2337511, 1374987977] 7 1119552
42042.db2 42042dk1 [1, 0, 0, -72521, -7522971] 1 159936 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 42042.db have rank $$1$$.

## Modular form 42042.2.a.db

sage: E.q_eigenform(10)
$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} + q^{11} + q^{12} - q^{13} - q^{15} + q^{16} - 3q^{17} + q^{18} - q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 