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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 15600b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 15600.s5 | 15600b1 | \([0, -1, 0, -232383, -46589238]\) | \(-5551350318708736/550618236675\) | \(-137654559168750000\) | \([2]\) | \(184320\) | \(2.0291\) | \(\Gamma_0(N)\)-optimal |
| 15600.s4 | 15600b2 | \([0, -1, 0, -3802508, -2852707488]\) | \(1520107298839022416/13013105625\) | \(52052422500000000\) | \([2, 2]\) | \(368640\) | \(2.3757\) | |
| 15600.s1 | 15600b3 | \([0, -1, 0, -60840008, -182634907488]\) | \(1556580279686303289604/114075\) | \(1825200000000\) | \([2]\) | \(737280\) | \(2.7223\) | |
| 15600.s3 | 15600b4 | \([0, -1, 0, -3887008, -2719197488]\) | \(405929061432816484/35083409765625\) | \(561334556250000000000\) | \([2, 2]\) | \(737280\) | \(2.7223\) | |
| 15600.s2 | 15600b5 | \([0, -1, 0, -13364008, 15704090512]\) | \(8248670337458940482/1446075439453125\) | \(46274414062500000000000\) | \([2]\) | \(1474560\) | \(3.0689\) | |
| 15600.s6 | 15600b6 | \([0, -1, 0, 4237992, -12599197488]\) | \(263059523447441758/2294739983908125\) | \(-73431679485060000000000\) | \([2]\) | \(1474560\) | \(3.0689\) |
Rank
sage: E.rank()
The elliptic curves in class 15600b have rank \(1\).
Complex multiplication
The elliptic curves in class 15600b do not have complex multiplication.Modular form 15600.2.a.b
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
