Properties

Label 321552.er
Number of curves $4$
Conductor $321552$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("er1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 321552.er

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
321552.er1 321552er3 \([0, 0, 0, -7683219, 8196251922]\) \(16798320881842096017/2132227789307\) \(6366798063226073088\) \([2]\) \(11010048\) \(2.6300\)  
321552.er2 321552er4 \([0, 0, 0, -3047859, -1964382030]\) \(1048626554636928177/48569076788309\) \(145026486184662061056\) \([2]\) \(11010048\) \(2.6300\)  
321552.er3 321552er2 \([0, 0, 0, -521379, 104805090]\) \(5249244962308257/1448621666569\) \(4325561118428368896\) \([2, 2]\) \(5505024\) \(2.2834\)  
321552.er4 321552er1 \([0, 0, 0, 84141, 10707282]\) \(22062729659823/29354283343\) \(-87651420393664512\) \([2]\) \(2752512\) \(1.9368\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 321552.er have rank \(0\).

Complex multiplication

The elliptic curves in class 321552.er do not have complex multiplication.

Modular form 321552.2.a.er

sage: E.q_eigenform(10)
 
\(q + 2q^{5} + q^{7} - q^{11} + 6q^{13} + 2q^{17} + 8q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.