# Properties

 Label 85800cp Number of curves $4$ Conductor $85800$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 85800cp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
85800.cy4 85800cp1 $$[0, 1, 0, -2283908, 1807694688]$$ $$-329381898333928144/162600887109375$$ $$-650403548437500000000$$ $$$$ $$4128768$$ $$2.6989$$ $$\Gamma_0(N)$$-optimal
85800.cy3 85800cp2 $$[0, 1, 0, -40096408, 97700194688]$$ $$445574312599094932036/61129333175625$$ $$978069330810000000000$$ $$[2, 2]$$ $$8257536$$ $$3.0455$$
85800.cy2 85800cp3 $$[0, 1, 0, -43671408, 79238894688]$$ $$287849398425814280018/81784533026485575$$ $$2617105056847538400000000$$ $$$$ $$16515072$$ $$3.3921$$
85800.cy1 85800cp4 $$[0, 1, 0, -641521408, 6253886494688]$$ $$912446049969377120252018/17177299425$$ $$549673581600000000$$ $$$$ $$16515072$$ $$3.3921$$

## Rank

sage: E.rank()

The elliptic curves in class 85800cp have rank $$1$$.

## Complex multiplication

The elliptic curves in class 85800cp do not have complex multiplication.

## Modular form 85800.2.a.cp

sage: E.q_eigenform(10)

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

## 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. 