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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 18720n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18720.bi3 | 18720n1 | \([0, 0, 0, -597, 5236]\) | \(504358336/38025\) | \(1774094400\) | \([2, 2]\) | \(6144\) | \(0.51993\) | \(\Gamma_0(N)\)-optimal |
18720.bi2 | 18720n2 | \([0, 0, 0, -1947, -26894]\) | \(2186875592/428415\) | \(159905041920\) | \([2]\) | \(12288\) | \(0.86651\) | |
18720.bi1 | 18720n3 | \([0, 0, 0, -9372, 349216]\) | \(30488290624/195\) | \(582266880\) | \([2]\) | \(12288\) | \(0.86651\) | |
18720.bi4 | 18720n4 | \([0, 0, 0, 573, 23254]\) | \(55742968/658125\) | \(-245643840000\) | \([2]\) | \(12288\) | \(0.86651\) |
Rank
sage: E.rank()
The elliptic curves in class 18720n have rank \(1\).
Complex multiplication
The elliptic curves in class 18720n do not have complex multiplication.Modular form 18720.2.a.n
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.