# Properties

 Label 693.d Number of curves 6 Conductor 693 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("693.d1")

sage: E.isogeny_class()

## Elliptic curves in class 693.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
693.d1 693d5 [1, -1, 0, -40671, 3167194]  1280
693.d2 693d3 [1, -1, 0, -2556, 49387] [2, 2] 640
693.d3 693d2 [1, -1, 0, -351, -1328] [2, 2] 320
693.d4 693d1 [1, -1, 0, -306, -1985]  160 $$\Gamma_0(N)$$-optimal
693.d5 693d6 [1, -1, 0, 279, 150880]  1280
693.d6 693d4 [1, -1, 0, 1134, -10535]  640

## Rank

sage: E.rank()

The elliptic curves in class 693.d have rank $$0$$.

## Modular form693.2.a.d

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} + 2q^{5} + q^{7} - 3q^{8} + 2q^{10} + q^{11} + 6q^{13} + q^{14} - q^{16} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 