# Properties

 Label 3648.o Number of curves $4$ Conductor $3648$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("o1")

E.isogeny_class()

## Elliptic curves in class 3648.o

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3648.o1 3648g3 $$[0, -1, 0, -6497, 203745]$$ $$115714886617/1539$$ $$403439616$$ $$$$ $$3072$$ $$0.79471$$
3648.o2 3648g2 $$[0, -1, 0, -417, 3105]$$ $$30664297/3249$$ $$851705856$$ $$[2, 2]$$ $$1536$$ $$0.44814$$
3648.o3 3648g1 $$[0, -1, 0, -97, -287]$$ $$389017/57$$ $$14942208$$ $$$$ $$768$$ $$0.10156$$ $$\Gamma_0(N)$$-optimal
3648.o4 3648g4 $$[0, -1, 0, 543, 14433]$$ $$67419143/390963$$ $$-102488604672$$ $$$$ $$3072$$ $$0.79471$$

## Rank

sage: E.rank()

The elliptic curves in class 3648.o have rank $$0$$.

## Complex multiplication

The elliptic curves in class 3648.o do not have complex multiplication.

## Modular form3648.2.a.o

sage: E.q_eigenform(10)

$$q - q^{3} + 2 q^{5} + q^{9} - 6 q^{13} - 2 q^{15} - 6 q^{17} + q^{19} + O(q^{20})$$

## 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 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. 