# Properties

 Label 135240do Number of curves $4$ Conductor $135240$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("do1")

sage: E.isogeny_class()

## Elliptic curves in class 135240do

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
135240.c4 135240do1 $$[0, -1, 0, -49996, 295156]$$ $$458891455696/264449745$$ $$7964735500673280$$ $$$$ $$737280$$ $$1.7407$$ $$\Gamma_0(N)$$-optimal
135240.c2 135240do2 $$[0, -1, 0, -568416, 164737980]$$ $$168591300897604/472410225$$ $$56912476734489600$$ $$[2, 2]$$ $$1474560$$ $$2.0873$$
135240.c1 135240do3 $$[0, -1, 0, -9088536, 10549060236]$$ $$344577854816148242/2716875$$ $$654617859840000$$ $$$$ $$2949120$$ $$2.4339$$
135240.c3 135240do4 $$[0, -1, 0, -343016, 296461740]$$ $$-18524646126002/146738831715$$ $$-35356010111873095680$$ $$$$ $$2949120$$ $$2.4339$$

## Rank

sage: E.rank()

The elliptic curves in class 135240do have rank $$0$$.

## Complex multiplication

The elliptic curves in class 135240do do not have complex multiplication.

## Modular form 135240.2.a.do

sage: E.q_eigenform(10)

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