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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 193200.cb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
193200.cb1 | 193200dq2 | \([0, -1, 0, -3671008, -630495488]\) | \(85486955243540761/46777901234400\) | \(2993785679001600000000\) | \([2]\) | \(7372800\) | \(2.8116\) | |
193200.cb2 | 193200dq1 | \([0, -1, 0, -2199008, 1247776512]\) | \(18374873741826841/136564270080\) | \(8740113285120000000\) | \([2]\) | \(3686400\) | \(2.4651\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 193200.cb have rank \(1\).
Complex multiplication
The elliptic curves in class 193200.cb do not have complex multiplication.Modular form 193200.2.a.cb
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.