Properties

Label 342.c
Number of curves $4$
Conductor $342$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 342.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
342.c1 342c3 \([1, -1, 0, -3852, 92988]\) \(8671983378625/82308\) \(60002532\) \([6]\) \(288\) \(0.65481\)  
342.c2 342c4 \([1, -1, 0, -3762, 97470]\) \(-8078253774625/846825858\) \(-617336050482\) \([6]\) \(576\) \(1.0014\)  
342.c3 342c1 \([1, -1, 0, -72, 0]\) \(57066625/32832\) \(23934528\) \([2]\) \(96\) \(0.10550\) \(\Gamma_0(N)\)-optimal
342.c4 342c2 \([1, -1, 0, 288, -216]\) \(3616805375/2105352\) \(-1534801608\) \([2]\) \(192\) \(0.45208\)  

Rank

sage: E.rank()
 

The elliptic curves in class 342.c have rank \(1\).

Complex multiplication

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

Modular form 342.2.a.c

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 4q^{7} - q^{8} - 4q^{13} + 4q^{14} + q^{16} - 6q^{17} + q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.