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Properties

Label 324870ci
Number of curves $4$
Conductor $324870$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("ci1")
 
E.isogeny_class()
 

Elliptic curves in class 324870ci

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
324870.ci4 324870ci1 \([1, 0, 1, -37903, 14918978]\) \(-51184652297689/788010612480\) \(-92708660547659520\) \([2]\) \(4325376\) \(1.9370\) \(\Gamma_0(N)\)-optimal
324870.ci3 324870ci2 \([1, 0, 1, -1170783, 485743906]\) \(1508565467598193369/6280737699600\) \(738922509620240400\) \([2, 2]\) \(8650752\) \(2.2836\)  
324870.ci1 324870ci3 \([1, 0, 1, -18713763, 31157890138]\) \(6160540455434488353049/107450752500\) \(12641473580872500\) \([2]\) \(17301504\) \(2.6302\)  
324870.ci2 324870ci4 \([1, 0, 1, -1753883, -49075414]\) \(5071506329733538969/2926108608384780\) \(344253751667860982220\) \([2]\) \(17301504\) \(2.6302\)  

Rank

sage: E.rank()
 

The elliptic curves in class 324870ci have rank \(2\).

Complex multiplication

The elliptic curves in class 324870ci do not have complex multiplication.

Modular form 324870.2.a.ci

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8} + q^{9} - q^{10} - 4 q^{11} + q^{12} - q^{13} + q^{15} + q^{16} - q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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.