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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 6720.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6720.o1 | 6720bl3 | \([0, -1, 0, -8961, 329505]\) | \(1214399773444/105\) | \(6881280\) | \([2]\) | \(4096\) | \(0.75257\) | |
6720.o2 | 6720bl2 | \([0, -1, 0, -561, 5265]\) | \(1193895376/11025\) | \(180633600\) | \([2, 2]\) | \(2048\) | \(0.40600\) | |
6720.o3 | 6720bl4 | \([0, -1, 0, -161, 12225]\) | \(-7086244/972405\) | \(-63727534080\) | \([2]\) | \(4096\) | \(0.75257\) | |
6720.o4 | 6720bl1 | \([0, -1, 0, -61, -35]\) | \(24918016/13125\) | \(13440000\) | \([2]\) | \(1024\) | \(0.059426\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6720.o have rank \(1\).
Complex multiplication
The elliptic curves in class 6720.o do not have complex multiplication.Modular form 6720.2.a.o
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.