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SageMath
E = EllipticCurve("em1")
E.isogeny_class()
Elliptic curves in class 100800em
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
100800.mb3 | 100800em1 | \([0, 0, 0, -6600, 205000]\) | \(2725888/21\) | \(244944000000\) | \([2]\) | \(131072\) | \(1.0144\) | \(\Gamma_0(N)\)-optimal |
100800.mb2 | 100800em2 | \([0, 0, 0, -11100, -110000]\) | \(810448/441\) | \(82301184000000\) | \([2, 2]\) | \(262144\) | \(1.3610\) | |
100800.mb4 | 100800em3 | \([0, 0, 0, 42900, -866000]\) | \(11696828/7203\) | \(-5377010688000000\) | \([2]\) | \(524288\) | \(1.7076\) | |
100800.mb1 | 100800em4 | \([0, 0, 0, -137100, -19514000]\) | \(381775972/567\) | \(423263232000000\) | \([2]\) | \(524288\) | \(1.7076\) |
Rank
sage: E.rank()
The elliptic curves in class 100800em have rank \(1\).
Complex multiplication
The elliptic curves in class 100800em do not have complex multiplication.Modular form 100800.2.a.em
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.