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SageMath
E = EllipticCurve("ej1")
E.isogeny_class()
Elliptic curves in class 76230ej
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
76230.ee8 | 76230ej1 | \([1, -1, 1, 1730398, 109268129]\) | \(443688652450511/260789760000\) | \(-336801621683197440000\) | \([4]\) | \(3317760\) | \(2.6282\) | \(\Gamma_0(N)\)-optimal |
76230.ee7 | 76230ej2 | \([1, -1, 1, -6981602, 882893729]\) | \(29141055407581489/16604321025600\) | \(21443948751555629006400\) | \([2, 2]\) | \(6635520\) | \(2.9748\) | |
76230.ee6 | 76230ej3 | \([1, -1, 1, -22053362, -43759694239]\) | \(-918468938249433649/109183593750000\) | \(-141007114068433593750000\) | \([4]\) | \(9953280\) | \(3.1775\) | |
76230.ee5 | 76230ej4 | \([1, -1, 1, -71668202, -232454609791]\) | \(31522423139920199089/164434491947880\) | \(212361879349475437455720\) | \([2]\) | \(13271040\) | \(3.3214\) | |
76230.ee4 | 76230ej5 | \([1, -1, 1, -81687002, 283627891649]\) | \(46676570542430835889/106752955783320\) | \(137868022990231084477080\) | \([2]\) | \(13271040\) | \(3.3214\) | |
76230.ee3 | 76230ej6 | \([1, -1, 1, -362365862, -2654909444239]\) | \(4074571110566294433649/48828650062500\) | \(63060637525228598062500\) | \([2, 2]\) | \(19906560\) | \(3.5241\) | |
76230.ee1 | 76230ej7 | \([1, -1, 1, -5797837112, -169919579126239]\) | \(16689299266861680229173649/2396798250\) | \(3095388168030254250\) | \([2]\) | \(39813120\) | \(3.8707\) | |
76230.ee2 | 76230ej8 | \([1, -1, 1, -371894612, -2507907512239]\) | \(4404531606962679693649/444872222400201750\) | \(574538225527704859262745750\) | \([2]\) | \(39813120\) | \(3.8707\) |
Rank
sage: E.rank()
The elliptic curves in class 76230ej have rank \(1\).
Complex multiplication
The elliptic curves in class 76230ej do not have complex multiplication.Modular form 76230.2.a.ej
sage: E.q_eigenform(10)
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.