# Properties

 Label 333270.dm Number of curves $4$ Conductor $333270$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("dm1")

sage: E.isogeny_class()

## Elliptic curves in class 333270.dm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
333270.dm1 333270dm3 [1, -1, 1, -1778333, 913226991] [2] 5767168
333270.dm2 333270dm2 [1, -1, 1, -111983, 14064531] [2, 2] 2883584
333270.dm3 333270dm1 [1, -1, 1, -16763, -523173] [2] 1441792 $$\Gamma_0(N)$$-optimal
333270.dm4 333270dm4 [1, -1, 1, 30847, 47372487] [2] 5767168

## Rank

sage: E.rank()

The elliptic curves in class 333270.dm have rank $$1$$.

## Complex multiplication

The elliptic curves in class 333270.dm do not have complex multiplication.

## Modular form 333270.2.a.dm

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.