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SageMath
E = EllipticCurve("hj1")
E.isogeny_class()
Elliptic curves in class 187200.hj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
187200.hj1 | 187200ef3 | \([0, 0, 0, -1540346700, -18008806606000]\) | \(1082883335268084577352/251301565117746585\) | \(93797806577068677358080000000\) | \([2]\) | \(165150720\) | \(4.2713\) | |
187200.hj2 | 187200ef2 | \([0, 0, 0, -1441481700, -21063537376000]\) | \(7099759044484031233216/577161945398025\) | \(26928067724490254400000000\) | \([2, 2]\) | \(82575360\) | \(3.9247\) | |
187200.hj3 | 187200ef1 | \([0, 0, 0, -1441453575, -21064400476000]\) | \(454357982636417669333824/3003024375\) | \(2189204769375000000\) | \([2]\) | \(41287680\) | \(3.5782\) | \(\Gamma_0(N)\)-optimal |
187200.hj4 | 187200ef4 | \([0, 0, 0, -1343066700, -24063029746000]\) | \(-717825640026599866952/254764560814329735\) | \(-95090362794826944929280000000\) | \([2]\) | \(165150720\) | \(4.2713\) |
Rank
sage: E.rank()
The elliptic curves in class 187200.hj have rank \(0\).
Complex multiplication
The elliptic curves in class 187200.hj do not have complex multiplication.Modular form 187200.2.a.hj
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.