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SageMath
E = EllipticCurve("nj1")
E.isogeny_class()
Elliptic curves in class 242550.nj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
242550.nj1 | 242550nj4 | \([1, -1, 1, -38808230, -92536241353]\) | \(4823468134087681/30382271150\) | \(40715149120401705468750\) | \([2]\) | \(31850496\) | \(3.1758\) | |
242550.nj2 | 242550nj2 | \([1, -1, 1, -2976980, 1892883647]\) | \(2177286259681/105875000\) | \(141882625951171875000\) | \([2]\) | \(10616832\) | \(2.6265\) | |
242550.nj3 | 242550nj3 | \([1, -1, 1, -992480, -3139808353]\) | \(-80677568161/3131816380\) | \(-4196933478075968437500\) | \([2]\) | \(15925248\) | \(2.8292\) | |
242550.nj4 | 242550nj1 | \([1, -1, 1, 110020, 114771647]\) | \(109902239/4312000\) | \(-5778492402375000000\) | \([2]\) | \(5308416\) | \(2.2799\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 242550.nj have rank \(1\).
Complex multiplication
The elliptic curves in class 242550.nj do not have complex multiplication.Modular form 242550.2.a.nj
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.