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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 412224.bb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
412224.bb1 | 412224bb3 | \([0, -1, 0, -274817, 55543233]\) | \(70049677573479176/6441\) | \(211058688\) | \([2]\) | \(1130496\) | \(1.4789\) | \(\Gamma_0(N)\)-optimal* |
412224.bb2 | 412224bb4 | \([0, -1, 0, -19457, 628065]\) | \(24861294286856/9293699577\) | \(304535947739136\) | \([2]\) | \(1130496\) | \(1.4789\) | |
412224.bb3 | 412224bb2 | \([0, -1, 0, -17177, 872025]\) | \(136845649240768/41486481\) | \(169928626176\) | \([2, 2]\) | \(565248\) | \(1.1324\) | \(\Gamma_0(N)\)-optimal* |
412224.bb4 | 412224bb1 | \([0, -1, 0, -932, 17538]\) | \(-1400416996672/1192828113\) | \(-76340999232\) | \([2]\) | \(282624\) | \(0.78579\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 412224.bb have rank \(1\).
Complex multiplication
The elliptic curves in class 412224.bb do not have complex multiplication.Modular form 412224.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.