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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 130130bz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 130130.bz4 | 130130bz1 | \([1, -1, 1, -4933, -1409859]\) | \(-2749884201/176619520\) | \(-852508688711680\) | \([2]\) | \(491520\) | \(1.5447\) | \(\Gamma_0(N)\)-optimal |
| 130130.bz3 | 130130bz2 | \([1, -1, 1, -221253, -39741763]\) | \(248158561089321/1859334400\) | \(8974652015929600\) | \([2, 2]\) | \(983040\) | \(1.8913\) | |
| 130130.bz2 | 130130bz3 | \([1, -1, 1, -369973, 20519581]\) | \(1160306142246441/634128110000\) | \(3060815268500990000\) | \([2]\) | \(1966080\) | \(2.2379\) | |
| 130130.bz1 | 130130bz4 | \([1, -1, 1, -3533653, -2555840803]\) | \(1010962818911303721/57392720\) | \(277023697430480\) | \([2]\) | \(1966080\) | \(2.2379\) |
Rank
sage: E.rank()
The elliptic curves in class 130130bz have rank \(1\).
Complex multiplication
The elliptic curves in class 130130bz do not have complex multiplication.Modular form 130130.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
