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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 13872be
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13872.bh2 | 13872be1 | \([0, 1, 0, -1181528, -494537196]\) | \(1845026709625/793152\) | \(78416941578190848\) | \([2]\) | \(165888\) | \(2.2019\) | \(\Gamma_0(N)\)-optimal |
| 13872.bh3 | 13872be2 | \([0, 1, 0, -996568, -654416620]\) | \(-1107111813625/1228691592\) | \(-121477644622314897408\) | \([2]\) | \(331776\) | \(2.5484\) | |
| 13872.bh1 | 13872be3 | \([0, 1, 0, -3470408, 1880154996]\) | \(46753267515625/11591221248\) | \(1145994865327579594752\) | \([2]\) | \(497664\) | \(2.7512\) | |
| 13872.bh4 | 13872be4 | \([0, 1, 0, 8367032, 11932509044]\) | \(655215969476375/1001033261568\) | \(-98969638594120286994432\) | \([2]\) | \(995328\) | \(3.0978\) |
Rank
sage: E.rank()
The elliptic curves in class 13872be have rank \(1\).
Complex multiplication
The elliptic curves in class 13872be do not have complex multiplication.Modular form 13872.2.a.be
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
