Properties

Label 1344b
Number of curves $4$
Conductor $1344$
CM no
Rank $1$
Graph

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

Elliptic curves in class 1344b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1344.c3 1344b1 \([0, -1, 0, -29, -51]\) \(2725888/21\) \(21504\) \([2]\) \(128\) \(-0.33961\) \(\Gamma_0(N)\)-optimal
1344.c2 1344b2 \([0, -1, 0, -49, 49]\) \(810448/441\) \(7225344\) \([2, 2]\) \(256\) \(0.0069590\)  
1344.c1 1344b3 \([0, -1, 0, -609, 5985]\) \(381775972/567\) \(37158912\) \([2]\) \(512\) \(0.35353\)  
1344.c4 1344b4 \([0, -1, 0, 191, 193]\) \(11696828/7203\) \(-472055808\) \([2]\) \(512\) \(0.35353\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 1344.2.a.b

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