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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 6720.bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6720.bp1 | 6720u5 | \([0, 1, 0, -1075201, -429482401]\) | \(524388516989299201/3150\) | \(825753600\) | \([2]\) | \(49152\) | \(1.7765\) | |
| 6720.bp2 | 6720u3 | \([0, 1, 0, -67201, -6727201]\) | \(128031684631201/9922500\) | \(2601123840000\) | \([2, 2]\) | \(24576\) | \(1.4299\) | |
| 6720.bp3 | 6720u6 | \([0, 1, 0, -62721, -7658145]\) | \(-104094944089921/35880468750\) | \(-9405849600000000\) | \([2]\) | \(49152\) | \(1.7765\) | |
| 6720.bp4 | 6720u4 | \([0, 1, 0, -23681, 1317855]\) | \(5602762882081/345888060\) | \(90672479600640\) | \([2]\) | \(24576\) | \(1.4299\) | |
| 6720.bp5 | 6720u2 | \([0, 1, 0, -4481, -91425]\) | \(37966934881/8643600\) | \(2265867878400\) | \([2, 2]\) | \(12288\) | \(1.0833\) | |
| 6720.bp6 | 6720u1 | \([0, 1, 0, 639, -8481]\) | \(109902239/188160\) | \(-49325015040\) | \([2]\) | \(6144\) | \(0.73676\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6720.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 6720.bp do not have complex multiplication.Modular form 6720.2.a.bp
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
