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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 39984p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
39984.dm3 | 39984p1 | \([0, 1, 0, -5847, 170148]\) | \(11745974272/357\) | \(672011088\) | \([2]\) | \(30720\) | \(0.79168\) | \(\Gamma_0(N)\)-optimal |
39984.dm2 | 39984p2 | \([0, 1, 0, -6092, 154860]\) | \(830321872/127449\) | \(3838527334656\) | \([2, 2]\) | \(61440\) | \(1.1383\) | |
39984.dm4 | 39984p3 | \([0, 1, 0, 10568, 867908]\) | \(1083360092/3306177\) | \(-398303659901952\) | \([2]\) | \(122880\) | \(1.4848\) | |
39984.dm1 | 39984p4 | \([0, 1, 0, -26672, -1532700]\) | \(17418812548/1753941\) | \(211301790422016\) | \([2]\) | \(122880\) | \(1.4848\) |
Rank
sage: E.rank()
The elliptic curves in class 39984p have rank \(1\).
Complex multiplication
The elliptic curves in class 39984p do not have complex multiplication.Modular form 39984.2.a.p
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.