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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 3990.bb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3990.bb1 | 3990bb5 | \([1, 0, 0, -355515, -81603333]\) | \(4969327007303723277361/1123462695162150\) | \(1123462695162150\) | \([2]\) | \(61440\) | \(1.8802\) | |
3990.bb2 | 3990bb3 | \([1, 0, 0, -24765, -966483]\) | \(1679731262160129361/570261564022500\) | \(570261564022500\) | \([2, 2]\) | \(30720\) | \(1.5336\) | |
3990.bb3 | 3990bb2 | \([1, 0, 0, -10185, 383625]\) | \(116844823575501841/3760263939600\) | \(3760263939600\) | \([2, 4]\) | \(15360\) | \(1.1870\) | |
3990.bb4 | 3990bb1 | \([1, 0, 0, -10105, 390137]\) | \(114113060120923921/124104960\) | \(124104960\) | \([4]\) | \(7680\) | \(0.84046\) | \(\Gamma_0(N)\)-optimal |
3990.bb5 | 3990bb4 | \([1, 0, 0, 3115, 1317285]\) | \(3342636501165359/751262567039460\) | \(-751262567039460\) | \([4]\) | \(30720\) | \(1.5336\) | |
3990.bb6 | 3990bb6 | \([1, 0, 0, 72705, -6678225]\) | \(42502666283088696719/43898058864843750\) | \(-43898058864843750\) | \([2]\) | \(61440\) | \(1.8802\) |
Rank
sage: E.rank()
The elliptic curves in class 3990.bb have rank \(0\).
Complex multiplication
The elliptic curves in class 3990.bb do not have complex multiplication.Modular form 3990.2.a.bb
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.