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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 50400bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
50400.eb3 | 50400bm1 | \([0, 0, 0, -11325, -218500]\) | \(220348864/99225\) | \(72335025000000\) | \([2, 2]\) | \(98304\) | \(1.3546\) | \(\Gamma_0(N)\)-optimal |
50400.eb4 | 50400bm2 | \([0, 0, 0, 39300, -1636000]\) | \(143877824/108045\) | \(-5040947520000000\) | \([2]\) | \(196608\) | \(1.7012\) | |
50400.eb2 | 50400bm3 | \([0, 0, 0, -90075, 10255250]\) | \(13858588808/229635\) | \(1339231320000000\) | \([2]\) | \(196608\) | \(1.7012\) | |
50400.eb1 | 50400bm4 | \([0, 0, 0, -153075, -23040250]\) | \(68017239368/39375\) | \(229635000000000\) | \([2]\) | \(196608\) | \(1.7012\) |
Rank
sage: E.rank()
The elliptic curves in class 50400bm have rank \(1\).
Complex multiplication
The elliptic curves in class 50400bm do not have complex multiplication.Modular form 50400.2.a.bm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.