Properties

Label 39270m
Number of curves $4$
Conductor $39270$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("m1")
 
E.isogeny_class()
 

Elliptic curves in class 39270m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
39270.m3 39270m1 \([1, 1, 0, -357, -2499]\) \(5053913144281/538784400\) \(538784400\) \([2]\) \(24576\) \(0.40979\) \(\Gamma_0(N)\)-optimal
39270.m2 39270m2 \([1, 1, 0, -1337, 15729]\) \(264621653112601/38553322500\) \(38553322500\) \([2, 2]\) \(49152\) \(0.75636\)  
39270.m4 39270m3 \([1, 1, 0, 2233, 89271]\) \(1230512292220679/4083466406250\) \(-4083466406250\) \([2]\) \(98304\) \(1.1029\)  
39270.m1 39270m4 \([1, 1, 0, -20587, 1128379]\) \(965019006588684601/26046023850\) \(26046023850\) \([2]\) \(98304\) \(1.1029\)  

Rank

sage: E.rank()
 

The elliptic curves in class 39270m have rank \(2\).

Complex multiplication

The elliptic curves in class 39270m do not have complex multiplication.

Modular form 39270.2.a.m

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - q^{10} - q^{11} - q^{12} - 2 q^{13} + q^{14} - q^{15} + q^{16} - q^{17} - q^{18} - 8 q^{19} + O(q^{20})\) 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.