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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 47190k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47190.n6 | 47190k1 | \([1, 1, 0, 1813, -11139]\) | \(371694959/249600\) | \(-442181625600\) | \([2]\) | \(81920\) | \(0.92330\) | \(\Gamma_0(N)\)-optimal |
| 47190.n5 | 47190k2 | \([1, 1, 0, -7867, -102131]\) | \(30400540561/15210000\) | \(26945442810000\) | \([2, 2]\) | \(163840\) | \(1.2699\) | |
| 47190.n3 | 47190k3 | \([1, 1, 0, -68367, 6782769]\) | \(19948814692561/231344100\) | \(409840185140100\) | \([2, 2]\) | \(327680\) | \(1.6164\) | |
| 47190.n2 | 47190k4 | \([1, 1, 0, -102247, -12616919]\) | \(66730743078481/60937500\) | \(107954498437500\) | \([2]\) | \(327680\) | \(1.6164\) | |
| 47190.n4 | 47190k5 | \([1, 1, 0, -13917, 17356959]\) | \(-168288035761/73415764890\) | \(-130060505864293290\) | \([2]\) | \(655360\) | \(1.9630\) | |
| 47190.n1 | 47190k6 | \([1, 1, 0, -1090817, 438052179]\) | \(81025909800741361/11088090\) | \(19643227808490\) | \([2]\) | \(655360\) | \(1.9630\) |
Rank
sage: E.rank()
The elliptic curves in class 47190k have rank \(1\).
Complex multiplication
The elliptic curves in class 47190k do not have complex multiplication.Modular form 47190.2.a.k
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.
