Properties

Label 369600pg
Number of curves $6$
Conductor $369600$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("pg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 369600pg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
369600.pg4 369600pg1 \([0, 1, 0, -424033, -106419937]\) \(2058561081361/12705\) \(52039680000000\) \([2]\) \(3145728\) \(1.8186\) \(\Gamma_0(N)\)-optimal
369600.pg3 369600pg2 \([0, 1, 0, -432033, -102203937]\) \(2177286259681/161417025\) \(661164134400000000\) \([2, 2]\) \(6291456\) \(2.1652\)  
369600.pg2 369600pg3 \([0, 1, 0, -1400033, 516348063]\) \(74093292126001/14707625625\) \(60242434560000000000\) \([2, 2]\) \(12582912\) \(2.5117\)  
369600.pg5 369600pg4 \([0, 1, 0, 407967, -450803937]\) \(1833318007919/22507682505\) \(-92191467540480000000\) \([2]\) \(12582912\) \(2.5117\)  
369600.pg1 369600pg5 \([0, 1, 0, -21200033, 37562148063]\) \(257260669489908001/14267882475\) \(58441246617600000000\) \([2]\) \(25165824\) \(2.8583\)  
369600.pg6 369600pg6 \([0, 1, 0, 2911967, 3073364063]\) \(666688497209279/1381398046875\) \(-5658206400000000000000\) \([2]\) \(25165824\) \(2.8583\)  

Rank

sage: E.rank()
 

The elliptic curves in class 369600pg have rank \(0\).

Complex multiplication

The elliptic curves in class 369600pg do not have complex multiplication.

Modular form 369600.2.a.pg

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{7} + q^{9} + q^{11} - 2 q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.