Show commands:
SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 4410.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4410.f1 | 4410h7 | \([1, -1, 0, -2844900, 1162675786]\) | \(29689921233686449/10380965400750\) | \(890335134657537990750\) | \([2]\) | \(221184\) | \(2.7213\) | |
| 4410.f2 | 4410h4 | \([1, -1, 0, -2540610, 1559308540]\) | \(21145699168383889/2593080\) | \(222398413042680\) | \([2]\) | \(73728\) | \(2.1720\) | |
| 4410.f3 | 4410h6 | \([1, -1, 0, -1191150, -486774464]\) | \(2179252305146449/66177562500\) | \(5675792832860062500\) | \([2, 2]\) | \(110592\) | \(2.3747\) | |
| 4410.f4 | 4410h3 | \([1, -1, 0, -1182330, -494534300]\) | \(2131200347946769/2058000\) | \(176506677018000\) | \([2]\) | \(55296\) | \(2.0281\) | |
| 4410.f5 | 4410h2 | \([1, -1, 0, -159210, 24258100]\) | \(5203798902289/57153600\) | \(4901842573185600\) | \([2, 2]\) | \(36864\) | \(1.8254\) | |
| 4410.f6 | 4410h5 | \([1, -1, 0, -35730, 60832876]\) | \(-58818484369/18600435000\) | \(-1595287158862635000\) | \([2]\) | \(73728\) | \(2.1720\) | |
| 4410.f7 | 4410h1 | \([1, -1, 0, -18090, -325004]\) | \(7633736209/3870720\) | \(331976639877120\) | \([2]\) | \(18432\) | \(1.4788\) | \(\Gamma_0(N)\)-optimal |
| 4410.f8 | 4410h8 | \([1, -1, 0, 321480, -1639701050]\) | \(42841933504271/13565917968750\) | \(-1163496161983886718750\) | \([2]\) | \(221184\) | \(2.7213\) |
Rank
sage: E.rank()
The elliptic curves in class 4410.f have rank \(1\).
Complex multiplication
The elliptic curves in class 4410.f do not have complex multiplication.Modular form 4410.2.a.f
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.
