Properties

Label 4056.i
Number of curves $6$
Conductor $4056$
CM no
Rank $1$
Graph

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

Elliptic curves in class 4056.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4056.i1 4056a5 \([0, -1, 0, -64952, -6349812]\) \(3065617154/9\) \(88967743488\) \([2]\) \(9216\) \(1.3303\)  
4056.i2 4056a4 \([0, -1, 0, -10872, 439932]\) \(28756228/3\) \(14827957248\) \([2]\) \(4608\) \(0.98370\)  
4056.i3 4056a3 \([0, -1, 0, -4112, -95460]\) \(1556068/81\) \(400354845696\) \([2, 2]\) \(4608\) \(0.98370\)  
4056.i4 4056a2 \([0, -1, 0, -732, 5940]\) \(35152/9\) \(11120967936\) \([2, 2]\) \(2304\) \(0.63712\)  
4056.i5 4056a1 \([0, -1, 0, 113, 532]\) \(2048/3\) \(-231686832\) \([2]\) \(1152\) \(0.29055\) \(\Gamma_0(N)\)-optimal
4056.i6 4056a6 \([0, -1, 0, 2648, -384788]\) \(207646/6561\) \(-64857485002752\) \([2]\) \(9216\) \(1.3303\)  

Rank

sage: E.rank()
 

The elliptic curves in class 4056.i have rank \(1\).

Complex multiplication

The elliptic curves in class 4056.i do not have complex multiplication.

Modular form 4056.2.a.i

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.