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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 8400.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8400.p1 | 8400bp5 | \([0, -1, 0, -6720008, 6707302512]\) | \(524388516989299201/3150\) | \(201600000000\) | \([2]\) | \(147456\) | \(2.2346\) | |
| 8400.p2 | 8400bp4 | \([0, -1, 0, -420008, 104902512]\) | \(128031684631201/9922500\) | \(635040000000000\) | \([2, 2]\) | \(73728\) | \(1.8881\) | |
| 8400.p3 | 8400bp6 | \([0, -1, 0, -392008, 119462512]\) | \(-104094944089921/35880468750\) | \(-2296350000000000000\) | \([2]\) | \(147456\) | \(2.2346\) | |
| 8400.p4 | 8400bp3 | \([0, -1, 0, -148008, -20665488]\) | \(5602762882081/345888060\) | \(22136835840000000\) | \([2]\) | \(73728\) | \(1.8881\) | |
| 8400.p5 | 8400bp2 | \([0, -1, 0, -28008, 1414512]\) | \(37966934881/8643600\) | \(553190400000000\) | \([2, 2]\) | \(36864\) | \(1.5415\) | |
| 8400.p6 | 8400bp1 | \([0, -1, 0, 3992, 134512]\) | \(109902239/188160\) | \(-12042240000000\) | \([2]\) | \(18432\) | \(1.1949\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8400.p have rank \(1\).
Complex multiplication
The elliptic curves in class 8400.p do not have complex multiplication.Modular form 8400.2.a.p
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.
