Properties

Label 342.b
Number of curves $4$
Conductor $342$
CM no
Rank $0$
Graph

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

Elliptic curves in class 342.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
342.b1 342f3 \([1, -1, 0, -787968, 269419360]\) \(74220219816682217473/16416\) \(11967264\) \([2]\) \(1920\) \(1.6507\)  
342.b2 342f2 \([1, -1, 0, -49248, 4218880]\) \(18120364883707393/269485056\) \(196454605824\) \([2, 2]\) \(960\) \(1.3042\)  
342.b3 342f4 \([1, -1, 0, -47808, 4476064]\) \(-16576888679672833/2216253521952\) \(-1615648817503008\) \([2]\) \(1920\) \(1.6507\)  
342.b4 342f1 \([1, -1, 0, -3168, 62464]\) \(4824238966273/537919488\) \(392143306752\) \([2]\) \(480\) \(0.95760\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 342.b have rank \(0\).

Complex multiplication

The elliptic curves in class 342.b do not have complex multiplication.

Modular form 342.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 2q^{5} - q^{8} + 2q^{10} + 4q^{11} + 2q^{13} + q^{16} + 6q^{17} - q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.