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SageMath
E = EllipticCurve("hc1")
E.isogeny_class()
Elliptic curves in class 446160.hc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
446160.hc1 | 446160hc4 | \([0, 1, 0, -16510680, -25827790572]\) | \(25176685646263969/57915000\) | \(1145014858690560000\) | \([2]\) | \(18579456\) | \(2.7086\) | |
446160.hc2 | 446160hc2 | \([0, 1, 0, -1043800, -394053100]\) | \(6361447449889/294465600\) | \(5821764437075558400\) | \([2, 2]\) | \(9289728\) | \(2.3621\) | |
446160.hc3 | 446160hc1 | \([0, 1, 0, -178520, 20935188]\) | \(31824875809/8785920\) | \(173702994859130880\) | \([2]\) | \(4644864\) | \(2.0155\) | \(\Gamma_0(N)\)-optimal* |
446160.hc4 | 446160hc3 | \([0, 1, 0, 578600, -1505721580]\) | \(1083523132511/50179392120\) | \(-992077174781317447680\) | \([2]\) | \(18579456\) | \(2.7086\) |
Rank
sage: E.rank()
The elliptic curves in class 446160.hc have rank \(1\).
Complex multiplication
The elliptic curves in class 446160.hc do not have complex multiplication.Modular form 446160.2.a.hc
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.