# Properties

 Label 101640.bf Number of curves $6$ Conductor $101640$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("101640.bf1")

sage: E.isogeny_class()

## Elliptic curves in class 101640.bf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
101640.bf1 101640p6 [0, -1, 0, -1268120, 550063692] [2] 1310720
101640.bf2 101640p4 [0, -1, 0, -82320, 7915932] [2, 2] 655360
101640.bf3 101640p2 [0, -1, 0, -21820, -1110668] [2, 2] 327680
101640.bf4 101640p1 [0, -1, 0, -21215, -1182300] [2] 163840 $$\Gamma_0(N)$$-optimal
101640.bf5 101640p3 [0, -1, 0, 29000, -5562500] [2] 655360
101640.bf6 101640p5 [0, -1, 0, 135480, 42502572] [2] 1310720

## Rank

sage: E.rank()

The elliptic curves in class 101640.bf have rank $$0$$.

## Modular form 101640.2.a.bf

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} + q^{7} + q^{9} + 2q^{13} - q^{15} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.