Properties

Label 1575.c
Number of curves $6$
Conductor $1575$
CM no
Rank $1$
Graph

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

Elliptic curves in class 1575.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1575.c1 1575g5 \([1, -1, 1, -176405, 28561722]\) \(53297461115137/147\) \(1674421875\) \([2]\) \(4096\) \(1.4282\)  
1575.c2 1575g4 \([1, -1, 1, -11030, 447972]\) \(13027640977/21609\) \(246140015625\) \([2, 2]\) \(2048\) \(1.0817\)  
1575.c3 1575g3 \([1, -1, 1, -8780, -312528]\) \(6570725617/45927\) \(523137234375\) \([2]\) \(2048\) \(1.0817\)  
1575.c4 1575g6 \([1, -1, 1, -7655, 724722]\) \(-4354703137/17294403\) \(-196994059171875\) \([2]\) \(4096\) \(1.4282\)  
1575.c5 1575g2 \([1, -1, 1, -905, 2472]\) \(7189057/3969\) \(45209390625\) \([2, 2]\) \(1024\) \(0.73508\)  
1575.c6 1575g1 \([1, -1, 1, 220, 222]\) \(103823/63\) \(-717609375\) \([2]\) \(512\) \(0.38851\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1575.c have rank \(1\).

Complex multiplication

The elliptic curves in class 1575.c do not have complex multiplication.

Modular form 1575.2.a.c

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + q^{7} + 3q^{8} - 4q^{11} + 2q^{13} - q^{14} - q^{16} - 6q^{17} + 4q^{19} + O(q^{20})\)  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.