Properties

 Label 33635b Number of curves $3$ Conductor $33635$ CM no Rank $1$ Graph

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Show commands for: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

Elliptic curves in class 33635b

sage: E.isogeny_class().curves

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

Rank

sage: E.rank()

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

Complex multiplication

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

Modular form 33635.2.a.b

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.