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SageMath
E = EllipticCurve("ed1")
E.isogeny_class()
Elliptic curves in class 127680.ed
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 127680.ed1 | 127680cg4 | \([0, 1, 0, -802816481, -8755597873281]\) | \(218289391029690300712901881/306514992000\) | \(80351066062848000\) | \([2]\) | \(20643840\) | \(3.4087\) | |
| 127680.ed2 | 127680cg3 | \([0, 1, 0, -52511201, -123387334785]\) | \(61085713691774408830201/10268551781250000000\) | \(2691839238144000000000000\) | \([2]\) | \(20643840\) | \(3.4087\) | |
| 127680.ed3 | 127680cg2 | \([0, 1, 0, -50176481, -136816177281]\) | \(53294746224000958661881/1997017344000000\) | \(523506114625536000000\) | \([2, 2]\) | \(10321920\) | \(3.0621\) | |
| 127680.ed4 | 127680cg1 | \([0, 1, 0, -2990561, -2345742465]\) | \(-11283450590382195961/2530373271552000\) | \(-663322170897727488000\) | \([2]\) | \(5160960\) | \(2.7155\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 127680.ed have rank \(1\).
Complex multiplication
The elliptic curves in class 127680.ed do not have complex multiplication.Modular form 127680.2.a.ed
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
