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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 14994ce
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
14994.ce2 | 14994ce1 | \([1, -1, 1, -112685, 14582229]\) | \(1845026709625/793152\) | \(68025570403392\) | \([2]\) | \(69120\) | \(1.6144\) | \(\Gamma_0(N)\)-optimal |
14994.ce3 | 14994ce2 | \([1, -1, 1, -95045, 19288581]\) | \(-1107111813625/1228691592\) | \(-105380111751154632\) | \([2]\) | \(138240\) | \(1.9610\) | |
14994.ce1 | 14994ce3 | \([1, -1, 1, -330980, -55327737]\) | \(46753267515625/11591221248\) | \(994134084093739008\) | \([2]\) | \(207360\) | \(2.1637\) | |
14994.ce4 | 14994ce4 | \([1, -1, 1, 797980, -351566841]\) | \(655215969476375/1001033261568\) | \(-85854739836665737728\) | \([2]\) | \(414720\) | \(2.5103\) |
Rank
sage: E.rank()
The elliptic curves in class 14994ce have rank \(0\).
Complex multiplication
The elliptic curves in class 14994ce do not have complex multiplication.Modular form 14994.2.a.ce
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.