Properties

Label 397800.cc
Number of curves $6$
Conductor $397800$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("cc1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 397800.cc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
397800.cc1 397800cc5 \([0, 0, 0, -43758075, 110552597750]\) \(397210600760070242/3536192675535\) \(82492302734880480000000\) \([2]\) \(34603008\) \(3.2204\)  
397800.cc2 397800cc3 \([0, 0, 0, -4743075, -1147347250]\) \(1011710313226084/536724738225\) \(6260357346656400000000\) \([2, 2]\) \(17301504\) \(2.8738\)  
397800.cc3 397800cc2 \([0, 0, 0, -3730575, -2770384750]\) \(1969080716416336/2472575625\) \(7210030522500000000\) \([2, 2]\) \(8650752\) \(2.5273\)  
397800.cc4 397800cc1 \([0, 0, 0, -3729450, -2772140875]\) \(31476797652269056/49725\) \(9062381250000\) \([2]\) \(4325376\) \(2.1807\) \(\Gamma_0(N)\)-optimal
397800.cc5 397800cc4 \([0, 0, 0, -2736075, -4281030250]\) \(-194204905090564/566398828125\) \(-6606475931250000000000\) \([2]\) \(17301504\) \(2.8738\)  
397800.cc6 397800cc6 \([0, 0, 0, 18071925, -8972892250]\) \(27980756504588158/17683545112935\) \(-412521740394547680000000\) \([2]\) \(34603008\) \(3.2204\)  

Rank

sage: E.rank()
 

The elliptic curves in class 397800.cc have rank \(0\).

Complex multiplication

The elliptic curves in class 397800.cc do not have complex multiplication.

Modular form 397800.2.a.cc

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.