Properties

Label 397800.cf
Number of curves $4$
Conductor $397800$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("cf1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 397800.cf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
397800.cf1 397800cf3 \([0, 0, 0, -7356675, 7680086750]\) \(1887517194957938/21849165\) \(509697321120000000\) \([2]\) \(9437184\) \(2.5489\)  
397800.cf2 397800cf2 \([0, 0, 0, -471675, 113471750]\) \(994958062276/98903025\) \(1153604883600000000\) \([2, 2]\) \(4718592\) \(2.2023\)  
397800.cf3 397800cf1 \([0, 0, 0, -107175, -11551750]\) \(46689225424/7249905\) \(21140722980000000\) \([2]\) \(2359296\) \(1.8557\) \(\Gamma_0(N)\)-optimal
397800.cf4 397800cf4 \([0, 0, 0, 581325, 548360750]\) \(931329171502/6107473125\) \(-142475133060000000000\) \([2]\) \(9437184\) \(2.5489\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 397800.2.a.cf

sage: E.q_eigenform(10)
 
\(q - 4q^{11} + q^{13} + q^{17} + 8q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.