# Properties

 Label 289800.cp Number of curves $2$ Conductor $289800$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 289800.cp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
289800.cp1 289800cp2 $$[0, 0, 0, -109875, -14017250]$$ $$12576878500/1127$$ $$13145328000000$$ $$$$ $$1105920$$ $$1.5573$$
289800.cp2 289800cp1 $$[0, 0, 0, -6375, -251750]$$ $$-9826000/3703$$ $$-10797948000000$$ $$$$ $$552960$$ $$1.2107$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 289800.cp have rank $$0$$.

## Complex multiplication

The elliptic curves in class 289800.cp do not have complex multiplication.

## Modular form 289800.2.a.cp

sage: E.q_eigenform(10)

$$q + q^{7} - 4q^{11} - 6q^{13} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 