Properties

Label 68400.cs
Number of curves $3$
Conductor $68400$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("cs1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 68400.cs

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68400.cs1 68400ec3 \([0, 0, 0, -2769600, -1774082000]\) \(-50357871050752/19\) \(-886464000000\) \([]\) \(559872\) \(2.0806\)  
68400.cs2 68400ec2 \([0, 0, 0, -33600, -2522000]\) \(-89915392/6859\) \(-320013504000000\) \([]\) \(186624\) \(1.5313\)  
68400.cs3 68400ec1 \([0, 0, 0, 2400, -2000]\) \(32768/19\) \(-886464000000\) \([]\) \(62208\) \(0.98200\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 68400.cs have rank \(1\).

Complex multiplication

The elliptic curves in class 68400.cs do not have complex multiplication.

Modular form 68400.2.a.cs

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.