# Properties

 Label 42350bu Number of curves $4$ Conductor $42350$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("42350.cd1")

sage: E.isogeny_class()

## Elliptic curves in class 42350bu

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
42350.cd4 42350bu1 [1, -1, 1, 6995, 356997]  122880 $$\Gamma_0(N)$$-optimal
42350.cd3 42350bu2 [1, -1, 1, -53505, 3865997] [2, 2] 245760
42350.cd2 42350bu3 [1, -1, 1, -265255, -49071503]  491520
42350.cd1 42350bu4 [1, -1, 1, -809755, 280653497]  491520

## Rank

sage: E.rank()

The elliptic curves in class 42350bu have rank $$1$$.

## Modular form 42350.2.a.cd

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - q^{7} + q^{8} - 3q^{9} - 6q^{13} - q^{14} + q^{16} + 2q^{17} - 3q^{18} + O(q^{20})$$

## 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. 