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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 31824bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31824.bg4 | 31824bk1 | \([0, 0, 0, -77619, 8323378]\) | \(17319700013617/25857\) | \(77208588288\) | \([2]\) | \(65536\) | \(1.3579\) | \(\Gamma_0(N)\)-optimal |
31824.bg3 | 31824bk2 | \([0, 0, 0, -78339, 8161090]\) | \(17806161424897/668584449\) | \(1996382467362816\) | \([2, 2]\) | \(131072\) | \(1.7045\) | |
31824.bg5 | 31824bk3 | \([0, 0, 0, 31821, 29289778]\) | \(1193377118543/124806800313\) | \(-372671108825812992\) | \([2]\) | \(262144\) | \(2.0510\) | |
31824.bg2 | 31824bk4 | \([0, 0, 0, -200019, -23354030]\) | \(296380748763217/92608836489\) | \(276528504014770176\) | \([2, 2]\) | \(262144\) | \(2.0510\) | |
31824.bg6 | 31824bk5 | \([0, 0, 0, 558141, -158154878]\) | \(6439735268725823/7345472585373\) | \(-21933463612362412032\) | \([2]\) | \(524288\) | \(2.3976\) | |
31824.bg1 | 31824bk6 | \([0, 0, 0, -2905059, -1905520862]\) | \(908031902324522977/161726530797\) | \(482912833335349248\) | \([2]\) | \(524288\) | \(2.3976\) |
Rank
sage: E.rank()
The elliptic curves in class 31824bk have rank \(0\).
Complex multiplication
The elliptic curves in class 31824bk do not have complex multiplication.Modular form 31824.2.a.bk
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.