# Properties

 Label 138600eo Number of curves $4$ Conductor $138600$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("eo1")

sage: E.isogeny_class()

## Elliptic curves in class 138600eo

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
138600.bd3 138600eo1 $$[0, 0, 0, -132068775, 584182224250]$$ $$87364831012240243408/1760913$$ $$5134822308000000$$ $$[2]$$ $$11796480$$ $$2.9980$$ $$\Gamma_0(N)$$-optimal
138600.bd2 138600eo2 $$[0, 0, 0, -132073275, 584140423750]$$ $$21843440425782779332/3100814593569$$ $$36167901419388816000000$$ $$[2, 2]$$ $$23592960$$ $$3.3446$$
138600.bd4 138600eo3 $$[0, 0, 0, -120166275, 693744358750]$$ $$-8226100326647904626/4152140742401883$$ $$-96861139238751126624000000$$ $$[2]$$ $$47185920$$ $$3.6912$$
138600.bd1 138600eo4 $$[0, 0, 0, -144052275, 471861256750]$$ $$14171198121996897746/4077720290568771$$ $$95125058938388289888000000$$ $$[2]$$ $$47185920$$ $$3.6912$$

## Rank

sage: E.rank()

The elliptic curves in class 138600eo have rank $$0$$.

## Complex multiplication

The elliptic curves in class 138600eo do not have complex multiplication.

## Modular form 138600.2.a.eo

sage: E.q_eigenform(10)

$$q - q^{7} - q^{11} + 6 q^{13} - 2 q^{17} - 8 q^{19} + O(q^{20})$$

## 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.