# Properties

 Label 18240.cp Number of curves $4$ Conductor $18240$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("cp1")

sage: E.isogeny_class()

## Elliptic curves in class 18240.cp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
18240.cp1 18240bn3 $$[0, 1, 0, -39745, -3058657]$$ $$26487576322129/44531250$$ $$11673600000000$$ $$$$ $$49152$$ $$1.4026$$
18240.cp2 18240bn2 $$[0, 1, 0, -3265, -16225]$$ $$14688124849/8122500$$ $$2129264640000$$ $$[2, 2]$$ $$24576$$ $$1.0561$$
18240.cp3 18240bn1 $$[0, 1, 0, -1985, 33183]$$ $$3301293169/22800$$ $$5976883200$$ $$$$ $$12288$$ $$0.70949$$ $$\Gamma_0(N)$$-optimal
18240.cp4 18240bn4 $$[0, 1, 0, 12735, -115425]$$ $$871257511151/527800050$$ $$-138359616307200$$ $$$$ $$49152$$ $$1.4026$$

## Rank

sage: E.rank()

The elliptic curves in class 18240.cp have rank $$0$$.

## Complex multiplication

The elliptic curves in class 18240.cp do not have complex multiplication.

## Modular form 18240.2.a.cp

sage: E.q_eigenform(10)

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