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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 172480.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
172480.q1 | 172480ep4 | \([0, 1, 0, -81899841, 277169852959]\) | \(1969902499564819009/63690429687500\) | \(1964275233536000000000000\) | \([2]\) | \(31850496\) | \(3.4353\) | |
172480.q2 | 172480ep2 | \([0, 1, 0, -11214401, -14330140385]\) | \(5057359576472449/51765560000\) | \(1596500572488335360000\) | \([2]\) | \(10616832\) | \(2.8860\) | |
172480.q3 | 172480ep1 | \([0, 1, 0, -175681, -551610081]\) | \(-19443408769/4249907200\) | \(-131071300645106483200\) | \([2]\) | \(5308416\) | \(2.5395\) | \(\Gamma_0(N)\)-optimal |
172480.q4 | 172480ep3 | \([0, 1, 0, 1580479, 14857991455]\) | \(14156681599871/3100231750000\) | \(-95614183710588928000000\) | \([2]\) | \(15925248\) | \(3.0888\) |
Rank
sage: E.rank()
The elliptic curves in class 172480.q have rank \(1\).
Complex multiplication
The elliptic curves in class 172480.q do not have complex multiplication.Modular form 172480.2.a.q
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.