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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 4719.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4719.k1 | 4719j3 | \([1, 0, 1, -830547, -291405149]\) | \(35765103905346817/1287\) | \(2279999007\) | \([2]\) | \(30720\) | \(1.7400\) | |
| 4719.k2 | 4719j5 | \([1, 0, 1, -364092, 81851779]\) | \(3013001140430737/108679952667\) | \(192533165626703187\) | \([2]\) | \(61440\) | \(2.0866\) | |
| 4719.k3 | 4719j4 | \([1, 0, 1, -57357, -3543245]\) | \(11779205551777/3763454409\) | \(6667189056262449\) | \([2, 2]\) | \(30720\) | \(1.7400\) | |
| 4719.k4 | 4719j2 | \([1, 0, 1, -51912, -4556015]\) | \(8732907467857/1656369\) | \(2934358722009\) | \([2, 2]\) | \(15360\) | \(1.3935\) | |
| 4719.k5 | 4719j1 | \([1, 0, 1, -2907, -86759]\) | \(-1532808577/938223\) | \(-1662119276103\) | \([2]\) | \(7680\) | \(1.0469\) | \(\Gamma_0(N)\)-optimal |
| 4719.k6 | 4719j6 | \([1, 0, 1, 162258, -24099209]\) | \(266679605718863/296110251723\) | \(-524577373652649603\) | \([2]\) | \(61440\) | \(2.0866\) |
Rank
sage: E.rank()
The elliptic curves in class 4719.k have rank \(1\).
Complex multiplication
The elliptic curves in class 4719.k do not have complex multiplication.Modular form 4719.2.a.k
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
