Properties

Label 106575r
Number of curves $6$
Conductor $106575$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 106575r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
106575.cg5 106575r1 \([1, 1, 0, -960425, -388284000]\) \(-53297461115137/4513839183\) \(-8297635406886984375\) \([2]\) \(2359296\) \(2.3752\) \(\Gamma_0(N)\)-optimal
106575.cg4 106575r2 \([1, 1, 0, -15666550, -23873965625]\) \(231331938231569617/1472026689\) \(2705976061471265625\) \([2, 2]\) \(4718592\) \(2.7218\)  
106575.cg3 106575r3 \([1, 1, 0, -15966675, -22912065000]\) \(244883173420511137/18418027974129\) \(33857227705129730015625\) \([2, 2]\) \(9437184\) \(3.0683\)  
106575.cg1 106575r4 \([1, 1, 0, -250664425, -1527625367750]\) \(947531277805646290177/38367\) \(70528737234375\) \([2]\) \(9437184\) \(3.0683\)  
106575.cg6 106575r5 \([1, 1, 0, 15289200, -101645614125]\) \(215015459663151503/2552757445339983\) \(-4692646260731307186984375\) \([2]\) \(18874368\) \(3.4149\)  
106575.cg2 106575r6 \([1, 1, 0, -52024550, 117389126625]\) \(8471112631466271697/1662662681263647\) \(3056415652937293842234375\) \([2]\) \(18874368\) \(3.4149\)  

Rank

sage: E.rank()
 

The elliptic curves in class 106575r have rank \(1\).

Complex multiplication

The elliptic curves in class 106575r do not have complex multiplication.

Modular form 106575.2.a.r

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.