Properties

Label 12480.cm
Number of curves $4$
Conductor $12480$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("cm1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 12480.cm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12480.cm1 12480x4 \([0, 1, 0, -33281, 2325855]\) \(31103978031362/195\) \(25559040\) \([2]\) \(24576\) \(1.0270\)  
12480.cm2 12480x3 \([0, 1, 0, -2881, 4895]\) \(20183398562/11567205\) \(1516136693760\) \([2]\) \(24576\) \(1.0270\)  
12480.cm3 12480x2 \([0, 1, 0, -2081, 35775]\) \(15214885924/38025\) \(2492006400\) \([2, 2]\) \(12288\) \(0.68038\)  
12480.cm4 12480x1 \([0, 1, 0, -81, 975]\) \(-3631696/24375\) \(-399360000\) \([2]\) \(6144\) \(0.33381\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 12480.cm have rank \(0\).

Complex multiplication

The elliptic curves in class 12480.cm do not have complex multiplication.

Modular form 12480.2.a.cm

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{5} + 4q^{7} + q^{9} - 4q^{11} - q^{13} - q^{15} + 6q^{17} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.