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SageMath
E = EllipticCurve("ga1")
E.isogeny_class()
Elliptic curves in class 127050ga
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
127050.gv8 | 127050ga1 | \([1, 1, 1, 4806662, -505870969]\) | \(443688652450511/260789760000\) | \(-7218827625240000000000\) | \([2]\) | \(9953280\) | \(2.8836\) | \(\Gamma_0(N)\)-optimal |
127050.gv7 | 127050ga2 | \([1, 1, 1, -19393338, -4087470969]\) | \(29141055407581489/16604321025600\) | \(459618243131765025000000\) | \([2, 2]\) | \(19906560\) | \(3.2302\) | |
127050.gv6 | 127050ga3 | \([1, 1, 1, -61259338, 202591177031]\) | \(-918468938249433649/109183593750000\) | \(-3022271820739746093750000\) | \([2]\) | \(29859840\) | \(3.4329\) | |
127050.gv5 | 127050ga4 | \([1, 1, 1, -199078338, 1076178749031]\) | \(31522423139920199089/164434491947880\) | \(4551652077963722510625000\) | \([2]\) | \(39813120\) | \(3.5768\) | |
127050.gv4 | 127050ga5 | \([1, 1, 1, -226908338, -1313092090969]\) | \(46676570542430835889/106752955783320\) | \(2954990204694596289375000\) | \([2]\) | \(39813120\) | \(3.5768\) | |
127050.gv3 | 127050ga6 | \([1, 1, 1, -1006571838, 12291247427031]\) | \(4074571110566294433649/48828650062500\) | \(1351608314583946289062500\) | \([2, 2]\) | \(59719680\) | \(3.7795\) | |
127050.gv1 | 127050ga7 | \([1, 1, 1, -16105103088, 786664718177031]\) | \(16689299266861680229173649/2396798250\) | \(66344911008878906250\) | \([2]\) | \(119439360\) | \(4.1261\) | |
127050.gv2 | 127050ga8 | \([1, 1, 1, -1033040588, 11610682927031]\) | \(4404531606962679693649/444872222400201750\) | \(12314348112305059569246093750\) | \([2]\) | \(119439360\) | \(4.1261\) |
Rank
sage: E.rank()
The elliptic curves in class 127050ga have rank \(0\).
Complex multiplication
The elliptic curves in class 127050ga do not have complex multiplication.Modular form 127050.2.a.ga
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.