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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 360640bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 360640.bd4 | 360640bd1 | \([0, 1, 0, -5773441, -5318596641]\) | \(690080604747409/3406760000\) | \(105067815171522560000\) | \([2]\) | \(23003136\) | \(2.6894\) | \(\Gamma_0(N)\)-optimal |
| 360640.bd3 | 360640bd2 | \([0, 1, 0, -8909441, 1093268959]\) | \(2535986675931409/1450751712200\) | \(44742603751717024563200\) | \([2]\) | \(46006272\) | \(3.0360\) | |
| 360640.bd2 | 360640bd3 | \([0, 1, 0, -33182081, 69721354975]\) | \(131010595463836369/7704101562500\) | \(237602038016000000000000\) | \([2]\) | \(69009408\) | \(3.2387\) | |
| 360640.bd1 | 360640bd4 | \([0, 1, 0, -523182081, 4605847354975]\) | \(513516182162686336369/1944885031250\) | \(59982159293063168000000\) | \([2]\) | \(138018816\) | \(3.5853\) |
Rank
sage: E.rank()
The elliptic curves in class 360640bd have rank \(1\).
Complex multiplication
The elliptic curves in class 360640bd do not have complex multiplication.Modular form 360640.2.a.bd
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.
