# Properties

 Label 3648.i Number of curves $4$ Conductor $3648$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

## Elliptic curves in class 3648.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3648.i1 3648b3 $$[0, -1, 0, -27393, -1735935]$$ $$8671983378625/82308$$ $$21576548352$$ $$$$ $$6912$$ $$1.1452$$
3648.i2 3648b4 $$[0, -1, 0, -26753, -1821567]$$ $$-8078253774625/846825858$$ $$-221990317719552$$ $$$$ $$13824$$ $$1.4918$$
3648.i3 3648b1 $$[0, -1, 0, -513, 513]$$ $$57066625/32832$$ $$8606711808$$ $$$$ $$2304$$ $$0.59592$$ $$\Gamma_0(N)$$-optimal
3648.i4 3648b2 $$[0, -1, 0, 2047, 2049]$$ $$3616805375/2105352$$ $$-551905394688$$ $$$$ $$4608$$ $$0.94249$$

## Rank

sage: E.rank()

The elliptic curves in class 3648.i have rank $$1$$.

## Complex multiplication

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

## Modular form3648.2.a.i

sage: E.q_eigenform(10)

$$q - q^{3} - 4 q^{7} + q^{9} + 4 q^{13} + 6 q^{17} - q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels. 