Properties

Label 75810co
Number of curves $8$
Conductor $75810$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("co1")
 
E.isogeny_class()
 

Elliptic curves in class 75810co

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75810.cn7 75810co1 \([1, 1, 1, -179605, -29360773]\) \(13619385906841/6048000\) \(284533488288000\) \([2]\) \(580608\) \(1.7320\) \(\Gamma_0(N)\)-optimal
75810.cn6 75810co2 \([1, 1, 1, -208485, -19322085]\) \(21302308926361/8930250000\) \(420131478800250000\) \([2, 2]\) \(1161216\) \(2.0786\)  
75810.cn5 75810co3 \([1, 1, 1, -531580, 113151197]\) \(353108405631241/86318776320\) \(4060942878816337920\) \([2]\) \(1741824\) \(2.2813\)  
75810.cn8 75810co4 \([1, 1, 1, 694015, -140979085]\) \(785793873833639/637994920500\) \(-30015033108447460500\) \([2]\) \(2322432\) \(2.4252\)  
75810.cn4 75810co5 \([1, 1, 1, -1573065, 745388547]\) \(9150443179640281/184570312500\) \(8683272958007812500\) \([2]\) \(2322432\) \(2.4252\)  
75810.cn2 75810co6 \([1, 1, 1, -7924860, 8582892765]\) \(1169975873419524361/108425318400\) \(5100964626833510400\) \([2, 2]\) \(3483648\) \(2.6279\)  
75810.cn3 75810co7 \([1, 1, 1, -7347260, 9887806685]\) \(-932348627918877961/358766164249920\) \(-16878470270128190579520\) \([2]\) \(6967296\) \(2.9745\)  
75810.cn1 75810co8 \([1, 1, 1, -126794940, 549489304797]\) \(4791901410190533590281/41160000\) \(1936408461960000\) \([2]\) \(6967296\) \(2.9745\)  

Rank

sage: E.rank()
 

The elliptic curves in class 75810co have rank \(1\).

Complex multiplication

The elliptic curves in class 75810co do not have complex multiplication.

Modular form 75810.2.a.co

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - q^{12} - 2 q^{13} + q^{14} - q^{15} + q^{16} - 6 q^{17} + q^{18} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.