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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 12870.g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12870.g1 | 12870m3 | \([1, -1, 0, -132887040, -589586574304]\) | \(355995140004443961140387841/2768480\) | \(2018221920\) | \([2]\) | \(860160\) | \(2.8075\) | |
| 12870.g2 | 12870m4 | \([1, -1, 0, -8324160, -9167111200]\) | \(87501897507774086005761/815991377947460000\) | \(594857714523698340000\) | \([2]\) | \(860160\) | \(2.8075\) | |
| 12870.g3 | 12870m2 | \([1, -1, 0, -8305440, -9210732544]\) | \(86912881496074271306241/7664481510400\) | \(5587407021081600\) | \([2, 2]\) | \(430080\) | \(2.4609\) | |
| 12870.g4 | 12870m1 | \([1, -1, 0, -517920, -144501760]\) | \(-21075830718885163521/199306463150080\) | \(-145294411636408320\) | \([2]\) | \(215040\) | \(2.1144\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12870.g have rank \(0\).
Complex multiplication
The elliptic curves in class 12870.g do not have complex multiplication.Modular form 12870.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
