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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 303600.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
303600.e1 | 303600e4 | \([0, -1, 0, -489533008, -932231271488]\) | \(202716771344720336812681/111444770370514561200\) | \(7132465303712931916800000000\) | \([2]\) | \(259522560\) | \(4.0352\) | |
303600.e2 | 303600e2 | \([0, -1, 0, -295133008, 1939445528512]\) | \(44421896119170051148681/319911052206240000\) | \(20474307341199360000000000\) | \([2, 2]\) | \(129761280\) | \(3.6886\) | |
303600.e3 | 303600e1 | \([0, -1, 0, -294621008, 1946550040512]\) | \(44191106172662624762761/2316725452800\) | \(148270428979200000000\) | \([2]\) | \(64880640\) | \(3.3420\) | \(\Gamma_0(N)\)-optimal |
303600.e4 | 303600e3 | \([0, -1, 0, -108925008, 4356425368512]\) | \(-2233194469464162213001/126809990183868750000\) | \(-8115839371767600000000000000\) | \([2]\) | \(259522560\) | \(4.0352\) |
Rank
sage: E.rank()
The elliptic curves in class 303600.e have rank \(1\).
Complex multiplication
The elliptic curves in class 303600.e do not have complex multiplication.Modular form 303600.2.a.e
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.