# Properties

 Label 9075.q Number of curves 4 Conductor 9075 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("9075.q1")
sage: E.isogeny_class()

## Elliptic curves in class 9075.q

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
9075.q1 9075k3 [1, 0, 1, -443226, -113501027] 2 92160
9075.q2 9075k2 [1, 0, 1, -34851, -789527] 4 46080
9075.q3 9075k1 [1, 0, 1, -19726, 1055723] 2 23040 $$\Gamma_0(N)$$-optimal
9075.q4 9075k4 [1, 0, 1, 131524, -6113527] 2 92160

## Rank

sage: E.rank()

The elliptic curves in class 9075.q have rank $$1$$.

## Modular form9075.2.a.q

sage: E.q_eigenform(10)
$$q + q^{2} + q^{3} - q^{4} + q^{6} + 4q^{7} - 3q^{8} + q^{9} - q^{12} - 2q^{13} + 4q^{14} - q^{16} - 2q^{17} + q^{18} + O(q^{20})$$

## 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 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. 