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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 106722.dd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 106722.dd1 | 106722df4 | \([1, -1, 0, -4909453236, -132401754214416]\) | \(86129359107301290313/9166294368\) | \(1392725987916942952250208\) | \([2]\) | \(88473600\) | \(4.0605\) | |
| 106722.dd2 | 106722df2 | \([1, -1, 0, -307600596, -2057959668528]\) | \(21184262604460873/216872764416\) | \(32951629409590441756394496\) | \([2, 2]\) | \(44236800\) | \(3.7139\) | |
| 106722.dd3 | 106722df3 | \([1, -1, 0, -77081076, -5071080314448]\) | \(-333345918055753/72923718045024\) | \(-11080023527434750844339682144\) | \([2]\) | \(88473600\) | \(4.0605\) | |
| 106722.dd4 | 106722df1 | \([1, -1, 0, -34392276, 25690904784]\) | \(29609739866953/15259926528\) | \(2318591940865842943623168\) | \([2]\) | \(22118400\) | \(3.3674\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 106722.dd have rank \(0\).
Complex multiplication
The elliptic curves in class 106722.dd do not have complex multiplication.Modular form 106722.2.a.dd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
