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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 32912bb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32912.y4 | 32912bb1 | \([0, -1, 0, -5848, 108528]\) | \(3048625/1088\) | \(7894869475328\) | \([2]\) | \(51840\) | \(1.1757\) | \(\Gamma_0(N)\)-optimal |
32912.y3 | 32912bb2 | \([0, -1, 0, -83288, 9277424]\) | \(8805624625/2312\) | \(16776597635072\) | \([2]\) | \(103680\) | \(1.5223\) | |
32912.y2 | 32912bb3 | \([0, -1, 0, -199448, -34212880]\) | \(120920208625/19652\) | \(142601079898112\) | \([2]\) | \(155520\) | \(1.7250\) | |
32912.y1 | 32912bb4 | \([0, -1, 0, -218808, -27150352]\) | \(159661140625/48275138\) | \(350299552769712128\) | \([2]\) | \(311040\) | \(2.0716\) |
Rank
sage: E.rank()
The elliptic curves in class 32912bb have rank \(0\).
Complex multiplication
The elliptic curves in class 32912bb do not have complex multiplication.Modular form 32912.2.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 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.