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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 84150.bz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84150.bz1 | 84150cd4 | \([1, -1, 0, -2692917, -1700241759]\) | \(189602977175292169/1402500\) | \(15975351562500\) | \([2]\) | \(1572864\) | \(2.1285\) | |
| 84150.bz2 | 84150cd3 | \([1, -1, 0, -235917, -3210759]\) | \(127483771761289/73369857660\) | \(835728534908437500\) | \([2]\) | \(1572864\) | \(2.1285\) | |
| 84150.bz3 | 84150cd2 | \([1, -1, 0, -168417, -26498259]\) | \(46380496070089/125888400\) | \(1433947556250000\) | \([2, 2]\) | \(786432\) | \(1.7819\) | |
| 84150.bz4 | 84150cd1 | \([1, -1, 0, -6417, -740259]\) | \(-2565726409/19388160\) | \(-220843260000000\) | \([2]\) | \(393216\) | \(1.4354\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 84150.bz have rank \(1\).
Complex multiplication
The elliptic curves in class 84150.bz do not have complex multiplication.Modular form 84150.2.a.bz
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
