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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 305184bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
305184.bc3 | 305184bc1 | \([0, -1, 0, -9922, 334480]\) | \(69934528/9801\) | \(15140628081216\) | \([2, 2]\) | \(655360\) | \(1.2553\) | \(\Gamma_0(N)\)-optimal |
305184.bc1 | 305184bc2 | \([0, -1, 0, -152977, 23080225]\) | \(4004529472/99\) | \(9787880779776\) | \([2]\) | \(1310720\) | \(1.6018\) | |
305184.bc4 | 305184bc3 | \([0, -1, 0, 16088, 1770232]\) | \(37259704/131769\) | \(-1628458664735232\) | \([2]\) | \(1310720\) | \(1.6018\) | |
305184.bc2 | 305184bc4 | \([0, -1, 0, -41712, -2933532]\) | \(649461896/72171\) | \(891920636057088\) | \([2]\) | \(1310720\) | \(1.6018\) |
Rank
sage: E.rank()
The elliptic curves in class 305184bc have rank \(1\).
Complex multiplication
The elliptic curves in class 305184bc do not have complex multiplication.Modular form 305184.2.a.bc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.