# Properties

 Label 1694f Number of curves 6 Conductor 1694 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("1694.e1")

sage: E.isogeny_class()

## Elliptic curves in class 1694f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1694.e5 1694f1 [1, 0, 0, -63, -395] [2] 480 $$\Gamma_0(N)$$-optimal
1694.e4 1694f2 [1, 0, 0, -1273, -17577] [2] 960
1694.e6 1694f3 [1, 0, 0, 542, 8196] [2] 1440
1694.e3 1694f4 [1, 0, 0, -4298, 88540] [2] 2880
1694.e2 1694f5 [1, 0, 0, -20633, 1142329] [2] 4320
1694.e1 1694f6 [1, 0, 0, -330393, 73068601] [2] 8640

## Rank

sage: E.rank()

The elliptic curves in class 1694f have rank $$1$$.

## Modular form1694.2.a.e

sage: E.q_eigenform(10)

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

## 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 Cremona numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.