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SageMath
E = EllipticCurve("bv1")
E.isogeny_class()
Elliptic curves in class 34848.bv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
34848.bv1 | 34848ca3 | \([0, 0, 0, -11979, -503118]\) | \(287496\) | \(661231600128\) | \([2]\) | \(46080\) | \(1.1309\) | \(-16\) | |
34848.bv2 | 34848ca4 | \([0, 0, 0, -11979, 503118]\) | \(287496\) | \(661231600128\) | \([2]\) | \(46080\) | \(1.1309\) | \(-16\) | |
34848.bv3 | 34848ca1 | \([0, 0, 0, -1089, 0]\) | \(1728\) | \(82653950016\) | \([2, 2]\) | \(23040\) | \(0.78429\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
34848.bv4 | 34848ca2 | \([0, 0, 0, 4356, 0]\) | \(1728\) | \(-5289852801024\) | \([2]\) | \(46080\) | \(1.1309\) | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 34848.bv have rank \(0\).
Complex multiplication
Each elliptic curve in class 34848.bv has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 34848.2.a.bv
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.