Properties

Label 102b
Number of curves $6$
Conductor $102$
CM no
Rank $0$
Graph

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

Elliptic curves in class 102b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
102.c5 102b1 \([1, 0, 0, -34, 68]\) \(4354703137/352512\) \(352512\) \([8]\) \(16\) \(-0.19264\) \(\Gamma_0(N)\)-optimal
102.c4 102b2 \([1, 0, 0, -114, -396]\) \(163936758817/30338064\) \(30338064\) \([2, 4]\) \(32\) \(0.15393\)  
102.c2 102b3 \([1, 0, 0, -1734, -27936]\) \(576615941610337/27060804\) \(27060804\) \([2, 2]\) \(64\) \(0.50050\)  
102.c6 102b4 \([1, 0, 0, 226, -2232]\) \(1276229915423/2927177028\) \(-2927177028\) \([4]\) \(64\) \(0.50050\)  
102.c1 102b5 \([1, 0, 0, -27744, -1781010]\) \(2361739090258884097/5202\) \(5202\) \([2]\) \(128\) \(0.84708\)  
102.c3 102b6 \([1, 0, 0, -1644, -30942]\) \(-491411892194497/125563633938\) \(-125563633938\) \([2]\) \(128\) \(0.84708\)  

Rank

sage: E.rank()
 

The elliptic curves in class 102b have rank \(0\).

Complex multiplication

The elliptic curves in class 102b do not have complex multiplication.

Modular form 102.2.a.b

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} - 2q^{5} + q^{6} + q^{8} + q^{9} - 2q^{10} - 4q^{11} + q^{12} - 2q^{13} - 2q^{15} + q^{16} + q^{17} + q^{18} + 4q^{19} + O(q^{20})\)  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 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.