# Properties

 Label 5775.q Number of curves $6$ Conductor $5775$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 5775.q

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5775.q1 5775c5 [1, 1, 0, -76230000, -256206911625] [2] 184320
5775.q2 5775c3 [1, 1, 0, -4764375, -4004721000] [2, 2] 92160
5775.q3 5775c6 [1, 1, 0, -4740750, -4046371875] [2] 184320
5775.q4 5775c4 [1, 1, 0, -636125, 102716250] [2] 92160
5775.q5 5775c2 [1, 1, 0, -299250, -62015625] [2, 2] 46080
5775.q6 5775c1 [1, 1, 0, 875, -2891000] [2] 23040 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 5775.q have rank $$1$$.

## Complex multiplication

The elliptic curves in class 5775.q do not have complex multiplication.

## Modular form5775.2.a.q

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} - q^{6} - q^{7} - 3q^{8} + q^{9} - q^{11} + q^{12} + 2q^{13} - q^{14} - q^{16} - 2q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.