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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 119130u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
119130.s6 | 119130u1 | \([1, 0, 1, 92047, -1013884]\) | \(1833318007919/1070530560\) | \(-50364053332623360\) | \([2]\) | \(1327104\) | \(1.8943\) | \(\Gamma_0(N)\)-optimal |
119130.s5 | 119130u2 | \([1, 0, 1, -370033, -8222332]\) | \(119102750067601/68309049600\) | \(3213659418704697600\) | \([2, 2]\) | \(2654208\) | \(2.2408\) | |
119130.s3 | 119130u3 | \([1, 0, 1, -3864513, 2911765156]\) | \(135670761487282321/643043610000\) | \(30252553153870410000\) | \([2, 2]\) | \(5308416\) | \(2.5874\) | |
119130.s2 | 119130u4 | \([1, 0, 1, -4268833, -3387702172]\) | \(182864522286982801/463015182960\) | \(21782957198729387760\) | \([2]\) | \(5308416\) | \(2.5874\) | |
119130.s4 | 119130u5 | \([1, 0, 1, -1879013, 5900339756]\) | \(-15595206456730321/310672490129100\) | \(-14615861000587313237100\) | \([2]\) | \(10616832\) | \(2.9340\) | |
119130.s1 | 119130u6 | \([1, 0, 1, -61761693, 186816367708]\) | \(553808571467029327441/12529687500\) | \(589470187092187500\) | \([2]\) | \(10616832\) | \(2.9340\) |
Rank
sage: E.rank()
The elliptic curves in class 119130u have rank \(1\).
Complex multiplication
The elliptic curves in class 119130u do not have complex multiplication.Modular form 119130.2.a.u
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.