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SageMath
E = EllipticCurve("qb1")
E.isogeny_class()
Elliptic curves in class 369600qb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 369600.qb4 | 369600qb1 | \([0, 1, 0, 123167, 9934463]\) | \(50447927375/39517632\) | \(-161864220672000000\) | \([2]\) | \(2654208\) | \(1.9901\) | \(\Gamma_0(N)\)-optimal |
| 369600.qb3 | 369600qb2 | \([0, 1, 0, -580833, 85262463]\) | \(5290763640625/2291573592\) | \(9386285432832000000\) | \([2]\) | \(5308416\) | \(2.3367\) | |
| 369600.qb2 | 369600qb3 | \([0, 1, 0, -1316833, -739153537]\) | \(-61653281712625/21875235228\) | \(-89600963493888000000\) | \([2]\) | \(7962624\) | \(2.5394\) | |
| 369600.qb1 | 369600qb4 | \([0, 1, 0, -22612833, -41393217537]\) | \(312196988566716625/25367712678\) | \(103906151129088000000\) | \([2]\) | \(15925248\) | \(2.8860\) |
Rank
sage: E.rank()
The elliptic curves in class 369600qb have rank \(1\).
Complex multiplication
The elliptic curves in class 369600qb do not have complex multiplication.Modular form 369600.2.a.qb
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
