# Properties

 Label 18240cm Number of curves $4$ Conductor $18240$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("18240.bo1")

sage: E.isogeny_class()

## Elliptic curves in class 18240cm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
18240.bo4 18240cm1 [0, 1, 0, -641, -10881]  18432 $$\Gamma_0(N)$$-optimal
18240.bo3 18240cm2 [0, 1, 0, -12161, -520065] [2, 2] 36864
18240.bo1 18240cm3 [0, 1, 0, -194561, -33096705]  73728
18240.bo2 18240cm4 [0, 1, 0, -14081, -346881]  73728

## Rank

sage: E.rank()

The elliptic curves in class 18240cm have rank $$1$$.

## Modular form 18240.2.a.bo

sage: E.q_eigenform(10)

$$q + q^{3} - q^{5} - 4q^{7} + q^{9} - 4q^{11} + 2q^{13} - q^{15} - 2q^{17} - q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 