Properties

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

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Show commands: SageMath
E = EllipticCurve("g1")
 
E.isogeny_class()
 

Elliptic curves in class 1344.g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1344.g1 1344a5 \([0, -1, 0, -50177, -4309503]\) \(53297461115137/147\) \(38535168\) \([2]\) \(2048\) \(1.1139\)  
1344.g2 1344a3 \([0, -1, 0, -3137, -66495]\) \(13027640977/21609\) \(5664669696\) \([2, 2]\) \(1024\) \(0.76735\)  
1344.g3 1344a4 \([0, -1, 0, -2497, 48577]\) \(6570725617/45927\) \(12039487488\) \([2]\) \(1024\) \(0.76735\)  
1344.g4 1344a6 \([0, -1, 0, -2177, -108927]\) \(-4354703137/17294403\) \(-4533623980032\) \([2]\) \(2048\) \(1.1139\)  
1344.g5 1344a2 \([0, -1, 0, -257, -255]\) \(7189057/3969\) \(1040449536\) \([2, 2]\) \(512\) \(0.42078\)  
1344.g6 1344a1 \([0, -1, 0, 63, -63]\) \(103823/63\) \(-16515072\) \([2]\) \(256\) \(0.074205\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 1344.g do not have complex multiplication.

Modular form 1344.2.a.g

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.