Properties

Label 12635c
Number of curves $4$
Conductor $12635$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 12635c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12635.c3 12635c1 \([1, -1, 1, -5122, 142104]\) \(315821241/665\) \(31285510865\) \([4]\) \(11520\) \(0.89891\) \(\Gamma_0(N)\)-optimal
12635.c2 12635c2 \([1, -1, 1, -6927, 34526]\) \(781229961/442225\) \(20804864725225\) \([2, 2]\) \(23040\) \(1.2455\)  
12635.c1 12635c3 \([1, -1, 1, -70102, -7091614]\) \(809818183161/4561235\) \(214587319023035\) \([2]\) \(46080\) \(1.5921\)  
12635.c4 12635c4 \([1, -1, 1, 27368, 254014]\) \(48188806119/28511875\) \(-1341366278336875\) \([2]\) \(46080\) \(1.5921\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12635c have rank \(1\).

Complex multiplication

The elliptic curves in class 12635c do not have complex multiplication.

Modular form 12635.2.a.c

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + q^{5} - q^{7} + 3 q^{8} - 3 q^{9} - q^{10} - 4 q^{11} + 2 q^{13} + q^{14} - q^{16} + 2 q^{17} + 3 q^{18} + O(q^{20})\) 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 Cremona 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 Cremona labels.