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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 64064bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 64064.z4 | 64064bn1 | \([0, 0, 0, -1004, 103248]\) | \(-426957777/17320303\) | \(-4540413509632\) | \([2]\) | \(77824\) | \(1.1088\) | \(\Gamma_0(N)\)-optimal |
| 64064.z3 | 64064bn2 | \([0, 0, 0, -39724, 3030480]\) | \(26444947540257/169338169\) | \(44390984974336\) | \([2, 2]\) | \(155648\) | \(1.4554\) | |
| 64064.z2 | 64064bn3 | \([0, 0, 0, -64364, -1178032]\) | \(112489728522417/62811265517\) | \(16465596387688448\) | \([2]\) | \(311296\) | \(1.8019\) | |
| 64064.z1 | 64064bn4 | \([0, 0, 0, -634604, 194581840]\) | \(107818231938348177/4463459\) | \(1170068996096\) | \([2]\) | \(311296\) | \(1.8019\) |
Rank
sage: E.rank()
The elliptic curves in class 64064bn have rank \(1\).
Complex multiplication
The elliptic curves in class 64064bn do not have complex multiplication.Modular form 64064.2.a.bn
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.
