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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 10830.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10830.g1 | 10830g4 | \([1, 1, 0, -19069263652, 1013550971091124]\) | \(16300610738133468173382620881/2228489100\) | \(104841233008397100\) | \([2]\) | \(8640000\) | \(4.0810\) | |
10830.g2 | 10830g3 | \([1, 1, 0, -1191828872, 15836364428016]\) | \(-3979640234041473454886161/1471455901872240\) | \(-69225939256229080243440\) | \([2]\) | \(4320000\) | \(3.7345\) | |
10830.g3 | 10830g2 | \([1, 1, 0, -31748152, 59307836224]\) | \(75224183150104868881/11219310000000000\) | \(527822323162110000000000\) | \([2]\) | \(1728000\) | \(3.2763\) | |
10830.g4 | 10830g1 | \([1, 1, 0, 3369928, 5064449856]\) | \(89962967236397039/287450726400000\) | \(-13523372667577958400000\) | \([2]\) | \(864000\) | \(2.9298\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 10830.g have rank \(1\).
Complex multiplication
The elliptic curves in class 10830.g do not have complex multiplication.Modular form 10830.2.a.g
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.