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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1050.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1050.a1 | 1050a7 | \([1, 1, 0, -161275, 15616375]\) | \(29689921233686449/10380965400750\) | \(162202584386718750\) | \([2]\) | \(13824\) | \(2.0037\) | |
| 1050.a2 | 1050a4 | \([1, 1, 0, -144025, 20978125]\) | \(21145699168383889/2593080\) | \(40516875000\) | \([2]\) | \(4608\) | \(1.4544\) | |
| 1050.a3 | 1050a6 | \([1, 1, 0, -67525, -6602375]\) | \(2179252305146449/66177562500\) | \(1034024414062500\) | \([2, 2]\) | \(6912\) | \(1.6572\) | |
| 1050.a4 | 1050a3 | \([1, 1, 0, -67025, -6706875]\) | \(2131200347946769/2058000\) | \(32156250000\) | \([2]\) | \(3456\) | \(1.3106\) | |
| 1050.a5 | 1050a2 | \([1, 1, 0, -9025, 323125]\) | \(5203798902289/57153600\) | \(893025000000\) | \([2, 2]\) | \(2304\) | \(1.1079\) | |
| 1050.a6 | 1050a5 | \([1, 1, 0, -2025, 820125]\) | \(-58818484369/18600435000\) | \(-290631796875000\) | \([2]\) | \(4608\) | \(1.4544\) | |
| 1050.a7 | 1050a1 | \([1, 1, 0, -1025, -4875]\) | \(7633736209/3870720\) | \(60480000000\) | \([2]\) | \(1152\) | \(0.76129\) | \(\Gamma_0(N)\)-optimal |
| 1050.a8 | 1050a8 | \([1, 1, 0, 18225, -22123125]\) | \(42841933504271/13565917968750\) | \(-211967468261718750\) | \([2]\) | \(13824\) | \(2.0037\) |
Rank
sage: E.rank()
The elliptic curves in class 1050.a have rank \(1\).
Complex multiplication
The elliptic curves in class 1050.a do not have complex multiplication.Modular form 1050.2.a.a
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
