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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 38088i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 38088.g5 | 38088i1 | \([0, 0, 0, 3174, 85169]\) | \(2048/3\) | \(-5180071827888\) | \([2]\) | \(45056\) | \(1.1251\) | \(\Gamma_0(N)\)-optimal |
| 38088.g4 | 38088i2 | \([0, 0, 0, -20631, 851690]\) | \(35152/9\) | \(248643447738624\) | \([2, 2]\) | \(90112\) | \(1.4717\) | |
| 38088.g3 | 38088i3 | \([0, 0, 0, -115851, -14478730]\) | \(1556068/81\) | \(8951164118590464\) | \([2, 2]\) | \(180224\) | \(1.8183\) | |
| 38088.g2 | 38088i4 | \([0, 0, 0, -306291, 65239454]\) | \(28756228/3\) | \(331524596984832\) | \([2]\) | \(180224\) | \(1.8183\) | |
| 38088.g6 | 38088i5 | \([0, 0, 0, 74589, -57403906]\) | \(207646/6561\) | \(-1450088587211655168\) | \([2]\) | \(360448\) | \(2.1648\) | |
| 38088.g1 | 38088i6 | \([0, 0, 0, -1829811, -952700434]\) | \(3065617154/9\) | \(1989147581908992\) | \([2]\) | \(360448\) | \(2.1648\) |
Rank
sage: E.rank()
The elliptic curves in class 38088i have rank \(1\).
Complex multiplication
The elliptic curves in class 38088i do not have complex multiplication.Modular form 38088.2.a.i
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
