# Properties

 Label 369600.wn Number of curves $4$ Conductor $369600$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("wn1")

sage: E.isogeny_class()

## Elliptic curves in class 369600.wn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
369600.wn1 369600wn4 $$[0, 1, 0, -493633, -133655137]$$ $$12990838708516/144375$$ $$147840000000000$$ $$$$ $$2359296$$ $$1.8724$$
369600.wn2 369600wn2 $$[0, 1, 0, -31633, -1985137]$$ $$13674725584/1334025$$ $$341510400000000$$ $$[2, 2]$$ $$1179648$$ $$1.5258$$
369600.wn3 369600wn1 $$[0, 1, 0, -7133, 195363]$$ $$2508888064/396165$$ $$6338640000000$$ $$$$ $$589824$$ $$1.1792$$ $$\Gamma_0(N)$$-optimal
369600.wn4 369600wn3 $$[0, 1, 0, 38367, -9475137]$$ $$6099383804/41507235$$ $$-42503408640000000$$ $$$$ $$2359296$$ $$1.8724$$

## Rank

sage: E.rank()

The elliptic curves in class 369600.wn have rank $$0$$.

## Complex multiplication

The elliptic curves in class 369600.wn do not have complex multiplication.

## Modular form 369600.2.a.wn

sage: E.q_eigenform(10)

$$q + q^{3} + q^{7} + q^{9} + q^{11} + 2 q^{13} - 6 q^{17} + O(q^{20})$$

## 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. 