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SageMath
E = EllipticCurve("cg1")
E.isogeny_class()
Elliptic curves in class 158400cg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
158400.h3 | 158400cg1 | \([0, 0, 0, -3073800, 2073287000]\) | \(275361373935616/148240125\) | \(1729072818000000000\) | \([2]\) | \(4718592\) | \(2.4486\) | \(\Gamma_0(N)\)-optimal |
158400.h2 | 158400cg2 | \([0, 0, 0, -3618300, 1288118000]\) | \(28071778927696/12404390625\) | \(2314956996000000000000\) | \([2, 2]\) | \(9437184\) | \(2.7952\) | |
158400.h4 | 158400cg3 | \([0, 0, 0, 12419700, 9595802000]\) | \(283811208976796/217529296875\) | \(-162384750000000000000000\) | \([2]\) | \(18874368\) | \(3.1418\) | |
158400.h1 | 158400cg4 | \([0, 0, 0, -28368300, -57270382000]\) | \(3382175663521924/59189241375\) | \(44184531929472000000000\) | \([2]\) | \(18874368\) | \(3.1418\) |
Rank
sage: E.rank()
The elliptic curves in class 158400cg have rank \(1\).
Complex multiplication
The elliptic curves in class 158400cg do not have complex multiplication.Modular form 158400.2.a.cg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.