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SageMath
E = EllipticCurve("kt1")
E.isogeny_class()
Elliptic curves in class 242550.kt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 242550.kt1 | 242550kt4 | \([1, -1, 1, -73368927680, 7649009018334947]\) | \(260744057755293612689909/8504954620259328\) | \(1424681576290372219974000000000\) | \([2]\) | \(737280000\) | \(4.8800\) | |
| 242550.kt2 | 242550kt3 | \([1, -1, 1, -4784607680, 108574540254947]\) | \(72313087342699809269/11447096545640448\) | \(1917525522328282589184000000000\) | \([2]\) | \(368640000\) | \(4.5334\) | |
| 242550.kt3 | 242550kt2 | \([1, -1, 1, -1298227055, -17838896930053]\) | \(1444540994277943589/15251205665388\) | \(2554759278307720156148437500\) | \([2]\) | \(147456000\) | \(4.0753\) | |
| 242550.kt4 | 242550kt1 | \([1, -1, 1, -1294919555, -17935125335053]\) | \(1433528304665250149/162339408\) | \(27193791620305406250000\) | \([2]\) | \(73728000\) | \(3.7287\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 242550.kt have rank \(1\).
Complex multiplication
The elliptic curves in class 242550.kt do not have complex multiplication.Modular form 242550.2.a.kt
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
