Properties

Label 137677b
Number of curves $3$
Conductor $137677$
CM no
Rank $1$
Graph

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

Elliptic curves in class 137677b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
137677.b3 137677b1 \([0, 1, 1, -12403, 523386]\) \(4096000/37\) \(1906253851357\) \([]\) \(151200\) \(1.1789\) \(\Gamma_0(N)\)-optimal
137677.b2 137677b2 \([0, 1, 1, -86823, -9564245]\) \(1404928000/50653\) \(2609661522507733\) \([]\) \(453600\) \(1.7282\)  
137677.b1 137677b3 \([0, 1, 1, -6970673, -7086024368]\) \(727057727488000/37\) \(1906253851357\) \([]\) \(1360800\) \(2.2775\)  

Rank

sage: E.rank()
 

The elliptic curves in class 137677b have rank \(1\).

Complex multiplication

The elliptic curves in class 137677b do not have complex multiplication.

Modular form 137677.2.a.b

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