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SageMath
E = EllipticCurve("oa1")
E.isogeny_class()
Elliptic curves in class 176400oa
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 176400.cg4 | 176400oa1 | \([0, 0, 0, -1933050, -1034445125]\) | \(37256083456/525\) | \(11256803381250000\) | \([2]\) | \(2359296\) | \(2.2202\) | \(\Gamma_0(N)\)-optimal |
| 176400.cg3 | 176400oa2 | \([0, 0, 0, -1988175, -972319250]\) | \(2533446736/275625\) | \(94557148402500000000\) | \([2, 2]\) | \(4718592\) | \(2.5667\) | |
| 176400.cg2 | 176400oa3 | \([0, 0, 0, -7500675, 6860943250]\) | \(34008619684/4862025\) | \(6671952391280400000000\) | \([2, 2]\) | \(9437184\) | \(2.9133\) | |
| 176400.cg5 | 176400oa4 | \([0, 0, 0, 2642325, -4829525750]\) | \(1486779836/8203125\) | \(-11256803381250000000000\) | \([2]\) | \(9437184\) | \(2.9133\) | |
| 176400.cg1 | 176400oa5 | \([0, 0, 0, -115545675, 478045188250]\) | \(62161150998242/1607445\) | \(4411658315867040000000\) | \([2]\) | \(18874368\) | \(3.2599\) | |
| 176400.cg6 | 176400oa6 | \([0, 0, 0, 12344325, 37005498250]\) | \(75798394558/259416045\) | \(-711971452953966240000000\) | \([2]\) | \(18874368\) | \(3.2599\) |
Rank
sage: E.rank()
The elliptic curves in class 176400oa have rank \(1\).
Complex multiplication
The elliptic curves in class 176400oa do not have complex multiplication.Modular form 176400.2.a.oa
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
