Properties

Label 3675.n
Number of curves $6$
Conductor $3675$
CM no
Rank $1$
Graph

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

Elliptic curves in class 3675.n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3675.n1 3675j5 \([1, 0, 1, -960426, 362199373]\) \(53297461115137/147\) \(270225046875\) \([2]\) \(24576\) \(1.8519\)  
3675.n2 3675j4 \([1, 0, 1, -60051, 5650873]\) \(13027640977/21609\) \(39723081890625\) \([2, 2]\) \(12288\) \(1.5053\)  
3675.n3 3675j3 \([1, 0, 1, -47801, -4002127]\) \(6570725617/45927\) \(84426025359375\) \([2]\) \(12288\) \(1.5053\)  
3675.n4 3675j6 \([1, 0, 1, -41676, 9178873]\) \(-4354703137/17294403\) \(-31791706539796875\) \([2]\) \(24576\) \(1.8519\)  
3675.n5 3675j2 \([1, 0, 1, -4926, 28123]\) \(7189057/3969\) \(7296076265625\) \([2, 2]\) \(6144\) \(1.1587\)  
3675.n6 3675j1 \([1, 0, 1, 1199, 3623]\) \(103823/63\) \(-115810734375\) \([2]\) \(3072\) \(0.81216\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 3675.n have rank \(1\).

Complex multiplication

The elliptic curves in class 3675.n do not have complex multiplication.

Modular form 3675.2.a.n

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} - q^{4} + q^{6} - 3q^{8} + q^{9} + 4q^{11} - q^{12} - 2q^{13} - q^{16} - 6q^{17} + q^{18} - 4q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.