Properties

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

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Show commands: SageMath
E = EllipticCurve("n1")
 
E.isogeny_class()
 

Elliptic curves in class 12705.n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12705.n1 12705m5 \([1, 0, 1, -368953203, 2727722241781]\) \(3135316978843283198764801/571725\) \(1012845712725\) \([2]\) \(921600\) \(3.0981\)  
12705.n2 12705m4 \([1, 0, 1, -23059578, 42619209631]\) \(765458482133960722801/326869475625\) \(579069215107700625\) \([2, 2]\) \(460800\) \(2.7515\)  
12705.n3 12705m6 \([1, 0, 1, -22945233, 43062822493]\) \(-754127868744065783521/15825714261328125\) \(-28036218182512714453125\) \([2]\) \(921600\) \(3.0981\)  
12705.n4 12705m3 \([1, 0, 1, -3078848, -1096801477]\) \(1821931919215868881/761147600816295\) \(1348419404849716386495\) \([2]\) \(460800\) \(2.7515\)  
12705.n5 12705m2 \([1, 0, 1, -1448373, 658894003]\) \(189674274234120481/3859869269025\) \(6837993862103198025\) \([2, 2]\) \(230400\) \(2.4049\)  
12705.n6 12705m1 \([1, 0, 1, 4232, 30787601]\) \(4733169839/231139696095\) \(-409478071153754295\) \([2]\) \(115200\) \(2.0584\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 12705.2.a.n

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.