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SageMath
sage: E = EllipticCurve("g1")
sage: E.isogeny_class()
Elliptic curves in class 6720.g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6720.g1 | 6720a3 | \([0, -1, 0, -56481, -5147775]\) | \(608119035935048/826875\) | \(27095040000\) | \([2]\) | \(12288\) | \(1.2761\) | |
| 6720.g2 | 6720a4 | \([0, -1, 0, -8961, 221121]\) | \(2428799546888/778248135\) | \(25501634887680\) | \([2]\) | \(12288\) | \(1.2761\) | |
| 6720.g3 | 6720a2 | \([0, -1, 0, -3561, -78039]\) | \(1219555693504/43758225\) | \(179233689600\) | \([2, 2]\) | \(6144\) | \(0.92955\) | |
| 6720.g4 | 6720a1 | \([0, -1, 0, 84, -4410]\) | \(1012048064/130203045\) | \(-8332994880\) | \([2]\) | \(3072\) | \(0.58297\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6720.g have rank \(1\).
Complex multiplication
The elliptic curves in class 6720.g do not have complex multiplication.Modular form 6720.2.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.
