Properties

Label 1344m
Number of curves $6$
Conductor $1344$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("1344.i1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1344m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1344.i5 1344m1 [0, -1, 0, -257, -3423] [2] 768 \(\Gamma_0(N)\)-optimal
1344.i4 1344m2 [0, -1, 0, -5377, -149855] [2, 2] 1536  
1344.i1 1344m3 [0, -1, 0, -86017, -9681503] [2] 3072  
1344.i3 1344m4 [0, -1, 0, -6657, -71775] [2, 2] 3072  
1344.i2 1344m5 [0, -1, 0, -58497, 5412897] [4] 6144  
1344.i6 1344m6 [0, -1, 0, 24703, -579807] [2] 6144  

Rank

sage: E.rank()
 

The elliptic curves in class 1344m have rank \(1\).

Modular form 1344.2.a.i

sage: E.q_eigenform(10)
 
\( q - q^{3} + 2q^{5} + q^{7} + q^{9} - 4q^{11} - 6q^{13} - 2q^{15} + 2q^{17} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.