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SageMath
E = EllipticCurve("ix1")
E.isogeny_class()
Elliptic curves in class 388080ix
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388080.ix4 | 388080ix1 | \([0, 0, 0, 70413, -58777166]\) | \(109902239/4312000\) | \(-1514797112328192000\) | \([2]\) | \(5308416\) | \(2.1683\) | \(\Gamma_0(N)\)-optimal |
| 388080.ix2 | 388080ix2 | \([0, 0, 0, -1905267, -968775374]\) | \(2177286259681/105875000\) | \(37193679097344000000\) | \([2]\) | \(10616832\) | \(2.5149\) | |
| 388080.ix3 | 388080ix3 | \([0, 0, 0, -635187, 1607708914]\) | \(-80677568161/3131816380\) | \(-1100200929676746670080\) | \([2]\) | \(15925248\) | \(2.7177\) | |
| 388080.ix1 | 388080ix4 | \([0, 0, 0, -24837267, 47383523026]\) | \(4823468134087681/30382271150\) | \(10673232051018584678400\) | \([2]\) | \(31850496\) | \(3.0642\) |
Rank
sage: E.rank()
The elliptic curves in class 388080ix have rank \(1\).
Complex multiplication
The elliptic curves in class 388080ix do not have complex multiplication.Modular form 388080.2.a.ix
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
