Properties

Label 12705m
Number of curves $6$
Conductor $12705$
CM no
Rank $0$
Graph

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

Elliptic curves in class 12705m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
12705.n6 12705m1 [1, 0, 1, 4232, 30787601] [2] 115200 \(\Gamma_0(N)\)-optimal
12705.n5 12705m2 [1, 0, 1, -1448373, 658894003] [2, 2] 230400  
12705.n4 12705m3 [1, 0, 1, -3078848, -1096801477] [2] 460800  
12705.n2 12705m4 [1, 0, 1, -23059578, 42619209631] [2, 2] 460800  
12705.n1 12705m5 [1, 0, 1, -368953203, 2727722241781] [2] 921600  
12705.n3 12705m6 [1, 0, 1, -22945233, 43062822493] [2] 921600  

Rank

sage: E.rank()
 

The elliptic curves in class 12705m have rank \(0\).

Complex multiplication

The elliptic curves in class 12705m do not have complex multiplication.

Modular form 12705.2.a.m

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.