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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 43890n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 43890.l3 | 43890n1 | \([1, 1, 0, -3762, -82764]\) | \(5890612465621801/560808864000\) | \(560808864000\) | \([2]\) | \(98304\) | \(0.99225\) | \(\Gamma_0(N)\)-optimal |
| 43890.l2 | 43890n2 | \([1, 1, 0, -13442, 503844]\) | \(268637023475732521/43342472250000\) | \(43342472250000\) | \([2, 2]\) | \(196608\) | \(1.3388\) | |
| 43890.l4 | 43890n3 | \([1, 1, 0, 24178, 2858856]\) | \(1562992036716078359/4410430664062500\) | \(-4410430664062500\) | \([2]\) | \(393216\) | \(1.6854\) | |
| 43890.l1 | 43890n4 | \([1, 1, 0, -205942, 35885344]\) | \(965965877509081052521/32918889118500\) | \(32918889118500\) | \([2]\) | \(393216\) | \(1.6854\) |
Rank
sage: E.rank()
The elliptic curves in class 43890n have rank \(2\).
Complex multiplication
The elliptic curves in class 43890n do not have complex multiplication.Modular form 43890.2.a.n
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.
