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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 12144.g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12144.g1 | 12144w5 | \([0, -1, 0, -1489664, 700307520]\) | \(89254274298475942657/17457\) | \(71503872\) | \([4]\) | \(65536\) | \(1.8083\) | |
| 12144.g2 | 12144w4 | \([0, -1, 0, -93104, 10965504]\) | \(21790813729717297/304746849\) | \(1248243093504\) | \([2, 4]\) | \(32768\) | \(1.4617\) | |
| 12144.g3 | 12144w6 | \([0, -1, 0, -90464, 11613888]\) | \(-19989223566735457/2584262514273\) | \(-10585139258462208\) | \([4]\) | \(65536\) | \(1.8083\) | |
| 12144.g4 | 12144w3 | \([0, -1, 0, -22544, -1122432]\) | \(309368403125137/44372288367\) | \(181748893151232\) | \([2]\) | \(32768\) | \(1.4617\) | |
| 12144.g5 | 12144w2 | \([0, -1, 0, -5984, 162624]\) | \(5786435182177/627352209\) | \(2569634648064\) | \([2, 2]\) | \(16384\) | \(1.1151\) | |
| 12144.g6 | 12144w1 | \([0, -1, 0, 496, 12288]\) | \(3288008303/18259263\) | \(-74789941248\) | \([2]\) | \(8192\) | \(0.76853\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12144.g have rank \(1\).
Complex multiplication
The elliptic curves in class 12144.g do not have complex multiplication.Modular form 12144.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
