Properties

Label 1344.r
Number of curves $4$
Conductor $1344$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 1344.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1344.r1 1344s3 \([0, 1, 0, -897, 10047]\) \(2438569736/21\) \(688128\) \([2]\) \(512\) \(0.28794\)  
1344.r2 1344s2 \([0, 1, 0, -57, 135]\) \(5088448/441\) \(1806336\) \([2, 2]\) \(256\) \(-0.058633\)  
1344.r3 1344s1 \([0, 1, 0, -12, -18]\) \(3241792/567\) \(36288\) \([2]\) \(128\) \(-0.40521\) \(\Gamma_0(N)\)-optimal
1344.r4 1344s4 \([0, 1, 0, 63, 735]\) \(830584/7203\) \(-236027904\) \([2]\) \(512\) \(0.28794\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1344.r have rank \(0\).

Complex multiplication

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

Modular form 1344.2.a.r

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.