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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 12705.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12705.g1 | 12705n3 | \([1, 0, 0, -13615, 610292]\) | \(157551496201/13125\) | \(23251738125\) | \([2]\) | \(23040\) | \(1.0330\) | |
12705.g2 | 12705n2 | \([1, 0, 0, -910, 8075]\) | \(47045881/11025\) | \(19531460025\) | \([2, 2]\) | \(11520\) | \(0.68641\) | |
12705.g3 | 12705n1 | \([1, 0, 0, -305, -1968]\) | \(1771561/105\) | \(186013905\) | \([2]\) | \(5760\) | \(0.33984\) | \(\Gamma_0(N)\)-optimal |
12705.g4 | 12705n4 | \([1, 0, 0, 2115, 51030]\) | \(590589719/972405\) | \(-1722674774205\) | \([2]\) | \(23040\) | \(1.0330\) |
Rank
sage: E.rank()
The elliptic curves in class 12705.g have rank \(0\).
Complex multiplication
The elliptic curves in class 12705.g do not have complex multiplication.Modular form 12705.2.a.g
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.