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SageMath
E = EllipticCurve("ea1")
E.isogeny_class()
Elliptic curves in class 76230.ea
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 76230.ea1 | 76230ek8 | \([1, -1, 1, -7025162, 4510720811]\) | \(29689921233686449/10380965400750\) | \(13406684302365873576750\) | \([2]\) | \(6635520\) | \(2.9473\) | |
| 76230.ea2 | 76230ek5 | \([1, -1, 1, -6273752, 6049948259]\) | \(21145699168383889/2593080\) | \(3348879761054520\) | \([2]\) | \(2211840\) | \(2.3980\) | |
| 76230.ea3 | 76230ek6 | \([1, -1, 1, -2941412, -1889332189]\) | \(2179252305146449/66177562500\) | \(85466202235245562500\) | \([2, 2]\) | \(3317760\) | \(2.6007\) | |
| 76230.ea4 | 76230ek3 | \([1, -1, 1, -2919632, -1919440861]\) | \(2131200347946769/2058000\) | \(2657841080202000\) | \([2]\) | \(1658880\) | \(2.2541\) | |
| 76230.ea5 | 76230ek2 | \([1, -1, 1, -393152, 94076579]\) | \(5203798902289/57153600\) | \(73812043713038400\) | \([2, 2]\) | \(1105920\) | \(2.0514\) | |
| 76230.ea6 | 76230ek4 | \([1, -1, 1, -88232, 236047331]\) | \(-58818484369/18600435000\) | \(-24021866011966515000\) | \([2]\) | \(2211840\) | \(2.3980\) | |
| 76230.ea7 | 76230ek1 | \([1, -1, 1, -44672, -1267549]\) | \(7633736209/3870720\) | \(4998910896967680\) | \([2]\) | \(552960\) | \(1.7048\) | \(\Gamma_0(N)\)-optimal |
| 76230.ea8 | 76230ek7 | \([1, -1, 1, 793858, -6362691541]\) | \(42841933504271/13565917968750\) | \(-17519948526722167968750\) | \([2]\) | \(6635520\) | \(2.9473\) |
Rank
sage: E.rank()
The elliptic curves in class 76230.ea have rank \(1\).
Complex multiplication
The elliptic curves in class 76230.ea do not have complex multiplication.Modular form 76230.2.a.ea
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.
