# Properties

 Label 60690.bt Number of curves $6$ Conductor $60690$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("60690.bt1")

sage: E.isogeny_class()

## Elliptic curves in class 60690.bt

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
60690.bt1 60690bu6 [1, 0, 0, -4855206, -4118151114] [2] 1310720
60690.bt2 60690bu4 [1, 0, 0, -303456, -64362564] [2, 2] 655360
60690.bt3 60690bu5 [1, 0, 0, -283226, -73308270] [2] 1310720
60690.bt4 60690bu3 [1, 0, 0, -106936, 12712580] [2] 655360
60690.bt5 60690bu2 [1, 0, 0, -20236, -864640] [2, 2] 327680
60690.bt6 60690bu1 [1, 0, 0, 2884, -83184] [2] 163840 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 60690.bt have rank $$1$$.

## Modular form 60690.2.a.bt

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - q^{7} + q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} - 2q^{13} - q^{14} - q^{15} + q^{16} + q^{18} - 4q^{19} + O(q^{20})$$

## 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.