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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 176610.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 176610.f1 | 176610do7 | \([1, 1, 0, -5425308, 3057792762]\) | \(29689921233686449/10380965400750\) | \(6174840314860210890750\) | \([2]\) | \(13934592\) | \(2.8827\) | |
| 176610.f2 | 176610do4 | \([1, 1, 0, -4845018, 4102773948]\) | \(21145699168383889/2593080\) | \(1542424457218680\) | \([2]\) | \(4644864\) | \(2.3334\) | |
| 176610.f3 | 176610do6 | \([1, 1, 0, -2271558, -1283659488]\) | \(2179252305146449/66177562500\) | \(39363957501935062500\) | \([2, 2]\) | \(6967296\) | \(2.5361\) | |
| 176610.f4 | 176610do3 | \([1, 1, 0, -2254738, -1304082332]\) | \(2131200347946769/2058000\) | \(1224146394618000\) | \([2]\) | \(3483648\) | \(2.1895\) | |
| 176610.f5 | 176610do2 | \([1, 1, 0, -303618, 63652788]\) | \(5203798902289/57153600\) | \(33996294159105600\) | \([2, 2]\) | \(2322432\) | \(1.9868\) | |
| 176610.f6 | 176610do5 | \([1, 1, 0, -68138, 160152492]\) | \(-58818484369/18600435000\) | \(-11063972518744635000\) | \([2]\) | \(4644864\) | \(2.3334\) | |
| 176610.f7 | 176610do1 | \([1, 1, 0, -34498, -882188]\) | \(7633736209/3870720\) | \(2302394525061120\) | \([2]\) | \(1161216\) | \(1.6402\) | \(\Gamma_0(N)\)-optimal |
| 176610.f8 | 176610do8 | \([1, 1, 0, 613072, -4317713322]\) | \(42841933504271/13565917968750\) | \(-8069324378585449218750\) | \([2]\) | \(13934592\) | \(2.8827\) |
Rank
sage: E.rank()
The elliptic curves in class 176610.f have rank \(0\).
Complex multiplication
The elliptic curves in class 176610.f do not have complex multiplication.Modular form 176610.2.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.
