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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 31920bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31920.bc3 | 31920bf1 | \([0, -1, 0, -210840, 37328112]\) | \(253060782505556761/41184460800\) | \(168691551436800\) | \([2]\) | \(147456\) | \(1.7389\) | \(\Gamma_0(N)\)-optimal |
| 31920.bc2 | 31920bf2 | \([0, -1, 0, -231320, 29660400]\) | \(334199035754662681/101099003040000\) | \(414101516451840000\) | \([2, 2]\) | \(294912\) | \(2.0855\) | |
| 31920.bc4 | 31920bf3 | \([0, -1, 0, 632680, 198313200]\) | \(6837784281928633319/8113766016106800\) | \(-33233985601973452800\) | \([4]\) | \(589824\) | \(2.4321\) | |
| 31920.bc1 | 31920bf4 | \([0, -1, 0, -1423000, -630053648]\) | \(77799851782095807001/3092322318750000\) | \(12666152217600000000\) | \([2]\) | \(589824\) | \(2.4321\) |
Rank
sage: E.rank()
The elliptic curves in class 31920bf have rank \(1\).
Complex multiplication
The elliptic curves in class 31920bf do not have complex multiplication.Modular form 31920.2.a.bf
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.
