Properties

Label 394944.il
Number of curves $6$
Conductor $394944$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("il1")
 
E.isogeny_class()
 

Elliptic curves in class 394944.il

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
394944.il1 394944il5 \([0, 1, 0, -3058446497, 65101747163583]\) \(6812873765474836663297/74052\) \(34390051434528768\) \([2]\) \(94371840\) \(3.6766\)  
394944.il2 394944il3 \([0, 1, 0, -191153057, 1017165320895]\) \(1663303207415737537/5483698704\) \(2546652088829724327936\) \([2, 2]\) \(47185920\) \(3.3300\)  
394944.il3 394944il6 \([0, 1, 0, -188520097, 1046549681087]\) \(-1595514095015181697/95635786040388\) \(-44413649879956404481032192\) \([2]\) \(94371840\) \(3.6766\)  
394944.il4 394944il2 \([0, 1, 0, -12111777, 15429359295]\) \(423108074414017/23284318464\) \(10813332652661749579776\) \([2, 2]\) \(23592960\) \(2.9835\)  
394944.il5 394944il1 \([0, 1, 0, -2199457, -951740737]\) \(2533811507137/625016832\) \(290260371089588748288\) \([2]\) \(11796480\) \(2.6369\) \(\Gamma_0(N)\)-optimal
394944.il6 394944il4 \([0, 1, 0, 8332383, 62242396863]\) \(137763859017023/3683199928848\) \(-1710493099399549217144832\) \([2]\) \(47185920\) \(3.3300\)  

Rank

sage: E.rank()
 

The elliptic curves in class 394944.il have rank \(1\).

Complex multiplication

The elliptic curves in class 394944.il do not have complex multiplication.

Modular form 394944.2.a.il

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.