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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 4680i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4680.s4 | 4680i1 | \([0, 0, 0, 1518, -1519]\) | \(33165879296/19278675\) | \(-224866465200\) | \([4]\) | \(3072\) | \(0.86767\) | \(\Gamma_0(N)\)-optimal |
| 4680.s3 | 4680i2 | \([0, 0, 0, -6087, -12166]\) | \(133649126224/77000625\) | \(14370164640000\) | \([2, 2]\) | \(6144\) | \(1.2142\) | |
| 4680.s1 | 4680i3 | \([0, 0, 0, -69267, -6999874]\) | \(49235161015876/137109375\) | \(102351600000000\) | \([2]\) | \(12288\) | \(1.5608\) | |
| 4680.s2 | 4680i4 | \([0, 0, 0, -64587, 6294134]\) | \(39914580075556/172718325\) | \(128933538739200\) | \([2]\) | \(12288\) | \(1.5608\) |
Rank
sage: E.rank()
The elliptic curves in class 4680i have rank \(0\).
Complex multiplication
The elliptic curves in class 4680i do not have complex multiplication.Modular form 4680.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
