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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 8470h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8470.j4 | 8470h1 | \([1, -1, 0, 280, 2800]\) | \(1367631/2800\) | \(-4960370800\) | \([2]\) | \(5120\) | \(0.54479\) | \(\Gamma_0(N)\)-optimal |
8470.j3 | 8470h2 | \([1, -1, 0, -2140, 31356]\) | \(611960049/122500\) | \(217016222500\) | \([2, 2]\) | \(10240\) | \(0.89136\) | |
8470.j2 | 8470h3 | \([1, -1, 0, -10610, -390450]\) | \(74565301329/5468750\) | \(9688224218750\) | \([2]\) | \(20480\) | \(1.2379\) | |
8470.j1 | 8470h4 | \([1, -1, 0, -32390, 2251706]\) | \(2121328796049/120050\) | \(212675898050\) | \([2]\) | \(20480\) | \(1.2379\) |
Rank
sage: E.rank()
The elliptic curves in class 8470h have rank \(1\).
Complex multiplication
The elliptic curves in class 8470h do not have complex multiplication.Modular form 8470.2.a.h
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.