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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 1734.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1734.j1 | 1734i5 | \([1, 1, 1, -8018022, -8742084111]\) | \(2361739090258884097/5202\) | \(125563633938\) | \([2]\) | \(36864\) | \(2.2637\) | |
| 1734.j2 | 1734i3 | \([1, 1, 1, -501132, -136748439]\) | \(576615941610337/27060804\) | \(653182023745476\) | \([2, 2]\) | \(18432\) | \(1.9171\) | |
| 1734.j3 | 1734i6 | \([1, 1, 1, -475122, -151542927]\) | \(-491411892194497/125563633938\) | \(-3030800878069216722\) | \([2]\) | \(36864\) | \(2.2637\) | |
| 1734.j4 | 1734i2 | \([1, 1, 1, -32952, -1912599]\) | \(163936758817/30338064\) | \(732287113126416\) | \([2, 2]\) | \(9216\) | \(1.5705\) | |
| 1734.j5 | 1734i1 | \([1, 1, 1, -9832, 343913]\) | \(4354703137/352512\) | \(8508782723328\) | \([4]\) | \(4608\) | \(1.2240\) | \(\Gamma_0(N)\)-optimal |
| 1734.j6 | 1734i4 | \([1, 1, 1, 65308, -11031127]\) | \(1276229915423/2927177028\) | \(-70654937488564932\) | \([2]\) | \(18432\) | \(1.9171\) |
Rank
sage: E.rank()
The elliptic curves in class 1734.j have rank \(0\).
Complex multiplication
The elliptic curves in class 1734.j do not have complex multiplication.Modular form 1734.2.a.j
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
