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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 47040.cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.cd1 | 47040bq6 | \([0, -1, 0, -52684865, 147207093825]\) | \(524388516989299201/3150\) | \(97149085286400\) | \([4]\) | \(2359296\) | \(2.7494\) | |
| 47040.cd2 | 47040bq4 | \([0, -1, 0, -3292865, 2300844225]\) | \(128031684631201/9922500\) | \(306019618652160000\) | \([2, 2]\) | \(1179648\) | \(2.4029\) | |
| 47040.cd3 | 47040bq5 | \([0, -1, 0, -3073345, 2620597057]\) | \(-104094944089921/35880468750\) | \(-1106588799590400000000\) | \([2]\) | \(2359296\) | \(2.7494\) | |
| 47040.cd4 | 47040bq3 | \([0, -1, 0, -1160385, -454345023]\) | \(5602762882081/345888060\) | \(10667526552535695360\) | \([2]\) | \(1179648\) | \(2.4029\) | |
| 47040.cd5 | 47040bq2 | \([0, -1, 0, -219585, 30919617]\) | \(37966934881/8643600\) | \(266577090025881600\) | \([2, 2]\) | \(589824\) | \(2.0563\) | |
| 47040.cd6 | 47040bq1 | \([0, -1, 0, 31295, 2971585]\) | \(109902239/188160\) | \(-5803038694440960\) | \([2]\) | \(294912\) | \(1.7097\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 47040.cd have rank \(1\).
Complex multiplication
The elliptic curves in class 47040.cd do not have complex multiplication.Modular form 47040.2.a.cd
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.
