Properties

Label 13357.b
Number of curves $3$
Conductor $13357$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 13357.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13357.b1 13357a3 \([0, -1, 1, -676273, 214283446]\) \(727057727488000/37\) \(1740697597\) \([]\) \(42768\) \(1.6943\)  
13357.b2 13357a2 \([0, -1, 1, -8423, 290949]\) \(1404928000/50653\) \(2383015010293\) \([]\) \(14256\) \(1.1450\)  
13357.b3 13357a1 \([0, -1, 1, -1203, -15540]\) \(4096000/37\) \(1740697597\) \([]\) \(4752\) \(0.59569\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 13357.b have rank \(0\).

Complex multiplication

The elliptic curves in class 13357.b do not have complex multiplication.

Modular form 13357.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{4} - q^{7} - 2 q^{9} + 3 q^{11} + 2 q^{12} + 4 q^{13} + 4 q^{16} + 6 q^{17} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.