Properties

Label 123210dh
Number of curves $6$
Conductor $123210$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("123210.dg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 123210dh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
123210.dg5 123210dh1 [1, -1, 1, -10411502, -44510052099] [4] 18911232 \(\Gamma_0(N)\)-optimal
123210.dg4 123210dh2 [1, -1, 1, -262745582, -1635829694211] [2, 2] 37822464  
123210.dg3 123210dh3 [1, -1, 1, -361313582, -296803127811] [2, 2] 75644928  
123210.dg1 123210dh4 [1, -1, 1, -4201522862, -104822341364739] [2] 75644928  
123210.dg6 123210dh5 [1, -1, 1, 1435088218, -2367695122851] [2] 151289856  
123210.dg2 123210dh6 [1, -1, 1, -3734803382, 87470606300829] [2] 151289856  

Rank

sage: E.rank()
 

The elliptic curves in class 123210dh have rank \(0\).

Modular form 123210.2.a.dg

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

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.