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Properties

Label 12705.b
Number of curves $4$
Conductor $12705$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 12705.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12705.b1 12705b3 \([1, 1, 1, -256946, 49482368]\) \(1058993490188089/13182390375\) \(23353408675125375\) \([2]\) \(138240\) \(1.9505\)  
12705.b2 12705b2 \([1, 1, 1, -30071, -793132]\) \(1697509118089/833765625\) \(1477066664390625\) \([2, 2]\) \(69120\) \(1.6040\)  
12705.b3 12705b1 \([1, 1, 1, -24626, -1496626]\) \(932288503609/779625\) \(1381153244625\) \([2]\) \(34560\) \(1.2574\) \(\Gamma_0(N)\)-optimal
12705.b4 12705b4 \([1, 1, 1, 109684, -5936116]\) \(82375335041831/56396484375\) \(-99909812255859375\) \([2]\) \(138240\) \(1.9505\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12705.b have rank \(2\).

Complex multiplication

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

Modular form 12705.2.a.b

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.