Properties

Label 2175.b
Number of curves $4$
Conductor $2175$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 2175.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2175.b1 2175c3 \([1, 1, 1, -6438, -198594]\) \(1888690601881/31827645\) \(497306953125\) \([2]\) \(4608\) \(1.0424\)  
2175.b2 2175c2 \([1, 1, 1, -813, 3906]\) \(3803721481/1703025\) \(26609765625\) \([2, 2]\) \(2304\) \(0.69580\)  
2175.b3 2175c1 \([1, 1, 1, -688, 6656]\) \(2305199161/1305\) \(20390625\) \([4]\) \(1152\) \(0.34922\) \(\Gamma_0(N)\)-optimal
2175.b4 2175c4 \([1, 1, 1, 2812, 32906]\) \(157376536199/118918125\) \(-1858095703125\) \([2]\) \(4608\) \(1.0424\)  

Rank

sage: E.rank()
 

The elliptic curves in class 2175.b have rank \(2\).

Complex multiplication

The elliptic curves in class 2175.b do not have complex multiplication.

Modular form 2175.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} + q^{6} - 4 q^{7} + 3 q^{8} + q^{9} - 4 q^{11} + q^{12} - 6 q^{13} + 4 q^{14} - q^{16} - 6 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.