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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 67760bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 67760.cd3 | 67760bt1 | \([0, -1, 0, -108456, 267570800]\) | \(-19443408769/4249907200\) | \(-30838660502074163200\) | \([2]\) | \(1658880\) | \(2.4189\) | \(\Gamma_0(N)\)-optimal |
| 67760.cd2 | 67760bt2 | \([0, -1, 0, -6923176, 6951448176]\) | \(5057359576472449/51765560000\) | \(375627150291599360000\) | \([2]\) | \(3317760\) | \(2.7654\) | |
| 67760.cd4 | 67760bt3 | \([0, -1, 0, 975704, -7207061904]\) | \(14156681599871/3100231750000\) | \(-22496254604336128000000\) | \([2]\) | \(4976640\) | \(2.9682\) | |
| 67760.cd1 | 67760bt4 | \([0, -1, 0, -50560616, -134439928720]\) | \(1969902499564819009/63690429687500\) | \(462157747436000000000000\) | \([2]\) | \(9953280\) | \(3.3148\) |
Rank
sage: E.rank()
The elliptic curves in class 67760bt have rank \(0\).
Complex multiplication
The elliptic curves in class 67760bt do not have complex multiplication.Modular form 67760.2.a.bt
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
