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Properties

Label 60690a
Number of curves $8$
Conductor $60690$
CM no
Rank $1$
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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 60690a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
60690.d7 60690a1 \([1, 1, 0, -143783, 20917173]\) \(13619385906841/6048000\) \(145984017312000\) \([2]\) \(442368\) \(1.6764\) \(\Gamma_0(N)\)-optimal
60690.d6 60690a2 \([1, 1, 0, -166903, 13708357]\) \(21302308926361/8930250000\) \(215554525562250000\) \([2, 2]\) \(884736\) \(2.0230\)  
60690.d5 60690a3 \([1, 1, 0, -425558, -81384492]\) \(353108405631241/86318776320\) \(2083525419419566080\) \([2]\) \(1327104\) \(2.2257\)  
60690.d8 60690a4 \([1, 1, 0, 555597, 101419857]\) \(785793873833639/637994920500\) \(-15399646415218264500\) \([2]\) \(1769472\) \(2.3696\)  
60690.d4 60690a5 \([1, 1, 0, -1259323, -534904967]\) \(9150443179640281/184570312500\) \(4455078653320312500\) \([2]\) \(1769472\) \(2.3696\)  
60690.d2 60690a6 \([1, 1, 0, -6344278, -6152807468]\) \(1169975873419524361/108425318400\) \(2617123604226969600\) \([2, 2]\) \(2654208\) \(2.5723\)  
60690.d3 60690a7 \([1, 1, 0, -5881878, -7087132908]\) \(-932348627918877961/358766164249920\) \(-8659743044447777244480\) \([2]\) \(5308416\) \(2.9189\)  
60690.d1 60690a8 \([1, 1, 0, -101506198, -393671178092]\) \(4791901410190533590281/41160000\) \(993502340040000\) \([2]\) \(5308416\) \(2.9189\)  

Rank

sage: E.rank()
 

The elliptic curves in class 60690a have rank \(1\).

Complex multiplication

The elliptic curves in class 60690a do not have complex multiplication.

Modular form 60690.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{7} - q^{8} + q^{9} + q^{10} - q^{12} + 2 q^{13} + q^{14} + q^{15} + q^{16} - q^{18} + 8 q^{19} + O(q^{20})\) 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 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.