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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 227430be
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 227430.ec6 | 227430be1 | \([1, -1, 1, 32422, -3152743]\) | \(109902239/188160\) | \(-6453219514371840\) | \([2]\) | \(1769472\) | \(1.7186\) | \(\Gamma_0(N)\)-optimal |
| 227430.ec5 | 227430be2 | \([1, -1, 1, -227498, -32471719]\) | \(37966934881/8643600\) | \(296444771441456400\) | \([2, 2]\) | \(3538944\) | \(2.0651\) | |
| 227430.ec4 | 227430be3 | \([1, -1, 1, -1202198, 479830601]\) | \(5602762882081/345888060\) | \(11862731603848946940\) | \([2]\) | \(7077888\) | \(2.4117\) | |
| 227430.ec2 | 227430be4 | \([1, -1, 1, -3411518, -2424307543]\) | \(128031684631201/9922500\) | \(340306497828202500\) | \([2, 2]\) | \(7077888\) | \(2.4117\) | |
| 227430.ec3 | 227430be5 | \([1, -1, 1, -3184088, -2761631719]\) | \(-104094944089921/35880468750\) | \(-1230572603753767968750\) | \([2]\) | \(14155776\) | \(2.7583\) | |
| 227430.ec1 | 227430be6 | \([1, -1, 1, -54583268, -155202684343]\) | \(524388516989299201/3150\) | \(108033808834350\) | \([2]\) | \(14155776\) | \(2.7583\) |
Rank
sage: E.rank()
The elliptic curves in class 227430be have rank \(1\).
Complex multiplication
The elliptic curves in class 227430be do not have complex multiplication.Modular form 227430.2.a.be
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.
