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SageMath
E = EllipticCurve("et1")
E.isogeny_class()
Elliptic curves in class 89280.et
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 89280.et1 | 89280fq6 | \([0, 0, 0, -177131532, -907385702384]\) | \(3216206300355197383681/57660\) | \(11018997596160\) | \([2]\) | \(6291456\) | \(2.9709\) | |
| 89280.et2 | 89280fq4 | \([0, 0, 0, -11070732, -14177871344]\) | \(785209010066844481/3324675600\) | \(635355401394585600\) | \([2, 2]\) | \(3145728\) | \(2.6243\) | |
| 89280.et3 | 89280fq5 | \([0, 0, 0, -10897932, -14641873904]\) | \(-749011598724977281/51173462246460\) | \(-9779400927522151464960\) | \([2]\) | \(6291456\) | \(2.9709\) | |
| 89280.et4 | 89280fq3 | \([0, 0, 0, -2131212, 933677584]\) | \(5601911201812801/1271193750000\) | \(242928908697600000000\) | \([2]\) | \(3145728\) | \(2.6243\) | |
| 89280.et5 | 89280fq2 | \([0, 0, 0, -702732, -214248944]\) | \(200828550012481/12454560000\) | \(2380103480770560000\) | \([2, 2]\) | \(1572864\) | \(2.2777\) | |
| 89280.et6 | 89280fq1 | \([0, 0, 0, 34548, -14003696]\) | \(23862997439/457113600\) | \(-87355769330073600\) | \([2]\) | \(786432\) | \(1.9311\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 89280.et have rank \(1\).
Complex multiplication
The elliptic curves in class 89280.et do not have complex multiplication.Modular form 89280.2.a.et
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.
