Properties

Label 75690t
Number of curves 8
Conductor 75690
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("75690.m1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 75690t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75690.m8 75690t1 [1, -1, 0, 11196, -1402880] [2] 387072 \(\Gamma_0(N)\)-optimal
75690.m6 75690t2 [1, -1, 0, -140184, -18266612] [2, 2] 774144  
75690.m7 75690t3 [1, -1, 0, -102339, 41263573] [2] 1161216  
75690.m5 75690t4 [1, -1, 0, -518634, 123954898] [2] 1548288  
75690.m4 75690t5 [1, -1, 0, -2183814, -1241583530] [2] 1548288  
75690.m3 75690t6 [1, -1, 0, -2524419, 1541499925] [2, 2] 2322432  
75690.m1 75690t7 [1, -1, 0, -40369419, 98735028925] [2] 4644864  
75690.m2 75690t8 [1, -1, 0, -3432699, 334032493] [2] 4644864  

Rank

sage: E.rank()
 

The elliptic curves in class 75690t have rank \(1\).

Modular form 75690.2.a.m

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{4} + q^{5} - 4q^{7} - q^{8} - q^{10} + 2q^{13} + 4q^{14} + q^{16} + 6q^{17} + 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.