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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 38115.u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
38115.u1 | 38115j4 | \([1, -1, 0, -22545045, -11667888050]\) | \(981281029968144361/522287841796875\) | \(674518018278803466796875\) | \([2]\) | \(4423680\) | \(3.2637\) | |
38115.u2 | 38115j2 | \([1, -1, 0, -17693550, -28610278889]\) | \(474334834335054841/607815140625\) | \(784973785190418140625\) | \([2, 2]\) | \(2211840\) | \(2.9172\) | |
38115.u3 | 38115j1 | \([1, -1, 0, -17688105, -28628790800]\) | \(473897054735271721/779625\) | \(1006860715331625\) | \([2]\) | \(1105920\) | \(2.5706\) | \(\Gamma_0(N)\)-optimal |
38115.u4 | 38115j3 | \([1, -1, 0, -12929175, -44367972764]\) | \(-185077034913624841/551466161890875\) | \(-712200884069433535882875\) | \([2]\) | \(4423680\) | \(3.2637\) |
Rank
sage: E.rank()
The elliptic curves in class 38115.u have rank \(1\).
Complex multiplication
The elliptic curves in class 38115.u do not have complex multiplication.Modular form 38115.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.