# Properties

 Label 33635.j Number of curves $3$ Conductor $33635$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("j1")

sage: E.isogeny_class()

## Elliptic curves in class 33635.j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
33635.j1 33635b3 $$[0, -1, 1, -126211, 18095692]$$ $$-250523582464/13671875$$ $$-12133839388671875$$ $$[]$$ $$181440$$ $$1.8445$$
33635.j2 33635b1 $$[0, -1, 1, -1281, -19158]$$ $$-262144/35$$ $$-31062628835$$ $$[]$$ $$20160$$ $$0.74584$$ $$\Gamma_0(N)$$-optimal
33635.j3 33635b2 $$[0, -1, 1, 8329, 47151]$$ $$71991296/42875$$ $$-38051720322875$$ $$[]$$ $$60480$$ $$1.2951$$

## Rank

sage: E.rank()

The elliptic curves in class 33635.j have rank $$1$$.

## Complex multiplication

The elliptic curves in class 33635.j do not have complex multiplication.

## Modular form 33635.2.a.j

sage: E.q_eigenform(10)

$$q - q^{3} - 2q^{4} - q^{5} + q^{7} - 2q^{9} + 3q^{11} + 2q^{12} - 5q^{13} + q^{15} + 4q^{16} - 3q^{17} + 2q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 