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Properties

Label 35557a
Number of curves $3$
Conductor $35557$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 35557a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
35557.c3 35557a1 \([0, -1, 1, -3203, -68169]\) \(4096000/37\) \(32837636197\) \([]\) \(20160\) \(0.84046\) \(\Gamma_0(N)\)-optimal
35557.c2 35557a2 \([0, -1, 1, -22423, 1258972]\) \(1404928000/50653\) \(44954723953693\) \([]\) \(60480\) \(1.3898\)  
35557.c1 35557a3 \([0, -1, 1, -1800273, 930327825]\) \(727057727488000/37\) \(32837636197\) \([]\) \(181440\) \(1.9391\)  

Rank

sage: E.rank()
 

The elliptic curves in class 35557a have rank \(2\).

Complex multiplication

The elliptic curves in class 35557a do not have complex multiplication.

Modular form 35557.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{4} - q^{7} - 2 q^{9} - 3 q^{11} + 2 q^{12} + 4 q^{13} + 4 q^{16} - 6 q^{17} + 2 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.