# Properties

 Label 72450dn Number of curves $6$ Conductor $72450$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("72450.dp1")

sage: E.isogeny_class()

## Elliptic curves in class 72450dn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
72450.dp5 72450dn1 [1, -1, 1, 28345, -3769153] [2] 524288 $$\Gamma_0(N)$$-optimal
72450.dp4 72450dn2 [1, -1, 1, -259655, -43513153] [2, 2] 1048576
72450.dp3 72450dn3 [1, -1, 1, -1141655, 427474847] [2, 2] 2097152
72450.dp2 72450dn4 [1, -1, 1, -3985655, -3061573153] [2] 2097152
72450.dp6 72450dn5 [1, -1, 1, 1409845, 2065537847] [2] 4194304
72450.dp1 72450dn6 [1, -1, 1, -17805155, 28922059847] [2] 4194304

## Rank

sage: E.rank()

The elliptic curves in class 72450dn have rank $$1$$.

## Modular form 72450.2.a.dp

sage: E.q_eigenform(10)

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

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.