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SageMath
E = EllipticCurve("gm1")
E.isogeny_class()
Elliptic curves in class 223440.gm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
223440.gm1 | 223440s4 | \([0, 1, 0, -967019720, -11574753414540]\) | \(207530301091125281552569/805586668007040\) | \(388204404344259579740160\) | \([2]\) | \(61931520\) | \(3.7395\) | |
223440.gm2 | 223440s3 | \([0, 1, 0, -183270600, 737277317748]\) | \(1412712966892699019449/330160465517040000\) | \(159101127096787922780160000\) | \([2]\) | \(61931520\) | \(3.7395\) | |
223440.gm3 | 223440s2 | \([0, 1, 0, -61342920, -175180668300]\) | \(52974743974734147769/3152005008998400\) | \(1518920651995761711513600\) | \([2, 2]\) | \(30965760\) | \(3.3929\) | |
223440.gm4 | 223440s1 | \([0, 1, 0, 2882360, -11303443852]\) | \(5495662324535111/117739817533440\) | \(-56737676464093931765760\) | \([2]\) | \(15482880\) | \(3.0464\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 223440.gm have rank \(1\).
Complex multiplication
The elliptic curves in class 223440.gm do not have complex multiplication.Modular form 223440.2.a.gm
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.