Properties

Label 12705.g
Number of curves $4$
Conductor $12705$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("g1")
 
E.isogeny_class()
 

Elliptic curves in class 12705.g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12705.g1 12705n3 \([1, 0, 0, -13615, 610292]\) \(157551496201/13125\) \(23251738125\) \([2]\) \(23040\) \(1.0330\)  
12705.g2 12705n2 \([1, 0, 0, -910, 8075]\) \(47045881/11025\) \(19531460025\) \([2, 2]\) \(11520\) \(0.68641\)  
12705.g3 12705n1 \([1, 0, 0, -305, -1968]\) \(1771561/105\) \(186013905\) \([2]\) \(5760\) \(0.33984\) \(\Gamma_0(N)\)-optimal
12705.g4 12705n4 \([1, 0, 0, 2115, 51030]\) \(590589719/972405\) \(-1722674774205\) \([2]\) \(23040\) \(1.0330\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 12705.2.a.g

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