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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 57222j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 57222.f5 | 57222j1 | \([1, -1, 0, -738738, -184932396]\) | \(2533811507137/625016832\) | \(10997976056341266432\) | \([2]\) | \(1179648\) | \(2.3641\) | \(\Gamma_0(N)\)-optimal |
| 57222.f4 | 57222j2 | \([1, -1, 0, -4068018, 3005183700]\) | \(423108074414017/23284318464\) | \(409717568942682257664\) | \([2, 2]\) | \(2359296\) | \(2.7107\) | |
| 57222.f6 | 57222j3 | \([1, -1, 0, 2798622, 12114468324]\) | \(137763859017023/3683199928848\) | \(-64810645976632130383248\) | \([2]\) | \(4718592\) | \(3.0573\) | |
| 57222.f2 | 57222j4 | \([1, -1, 0, -64203138, 198023377860]\) | \(1663303207415737537/5483698704\) | \(96492740609554709904\) | \([2, 2]\) | \(4718592\) | \(3.0573\) | |
| 57222.f3 | 57222j5 | \([1, -1, 0, -63318798, 203742758376]\) | \(-1595514095015181697/95635786040388\) | \(-1682834815241525457706788\) | \([2]\) | \(9437184\) | \(3.4038\) | |
| 57222.f1 | 57222j6 | \([1, -1, 0, -1027249398, 12672746802144]\) | \(6812873765474836663297/74052\) | \(1303040304239652\) | \([2]\) | \(9437184\) | \(3.4038\) |
Rank
sage: E.rank()
The elliptic curves in class 57222j have rank \(0\).
Complex multiplication
The elliptic curves in class 57222j do not have complex multiplication.Modular form 57222.2.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 Cremona 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 Cremona labels.
