Properties

Label 392.d
Number of curves $4$
Conductor $392$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 392.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
392.d1 392a3 \([0, 0, 0, -14651, -682570]\) \(1443468546/7\) \(1686616064\) \([2]\) \(384\) \(0.97092\)  
392.d2 392a4 \([0, 0, 0, -2891, 47334]\) \(11090466/2401\) \(578509309952\) \([2]\) \(384\) \(0.97092\)  
392.d3 392a2 \([0, 0, 0, -931, -10290]\) \(740772/49\) \(5903156224\) \([2, 2]\) \(192\) \(0.62435\)  
392.d4 392a1 \([0, 0, 0, 49, -686]\) \(432/7\) \(-210827008\) \([4]\) \(96\) \(0.27778\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 392.d have rank \(1\).

Complex multiplication

The elliptic curves in class 392.d do not have complex multiplication.

Modular form 392.2.a.d

sage: E.q_eigenform(10)
 
\(q - 2q^{5} - 3q^{9} - 4q^{11} - 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.