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SageMath
E = EllipticCurve("cy1")
E.isogeny_class()
Elliptic curves in class 72450.cy
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72450.cy1 | 72450df8 | \([1, -1, 1, -57127278605, 4930996687646397]\) | \(1810117493172631097464564372609/125368453502655029296875000\) | \(1428025040678679943084716796875000\) | \([2]\) | \(382205952\) | \(5.1098\) | |
| 72450.cy2 | 72450df6 | \([1, -1, 1, -56141751605, 5120073984704397]\) | \(1718043013877225552292911401729/9180538178765625000000\) | \(104572067692502197265625000000\) | \([2, 2]\) | \(191102976\) | \(4.7632\) | |
| 72450.cy3 | 72450df3 | \([1, -1, 1, -56141679605, 5120087774000397]\) | \(1718036403880129446396978632449/49057344000000\) | \(558793809000000000000\) | \([2]\) | \(95551488\) | \(4.4166\) | |
| 72450.cy4 | 72450df7 | \([1, -1, 1, -55157376605, 5308268765954397]\) | \(-1629247127728109256861881401729/125809119536174660320875000\) | \(-1433044502216739490217466796875000\) | \([2]\) | \(382205952\) | \(5.1098\) | |
| 72450.cy5 | 72450df5 | \([1, -1, 1, -10646319605, -421359643983603]\) | \(11715873038622856702991202049/46415372499833400000000\) | \(528700102380914821875000000000\) | \([2]\) | \(127401984\) | \(4.5604\) | |
| 72450.cy6 | 72450df2 | \([1, -1, 1, -988527605, 473394992397]\) | \(9378698233516887309850369/5418996968417034240000\) | \(61725762343375280640000000000\) | \([2, 2]\) | \(63700992\) | \(4.2139\) | |
| 72450.cy7 | 72450df1 | \([1, -1, 1, -693615605, 7012773680397]\) | \(3239908336204082689644289/9880281924658790400\) | \(112542586298066534400000000\) | \([2]\) | \(31850496\) | \(3.8673\) | \(\Gamma_0(N)\)-optimal |
| 72450.cy8 | 72450df4 | \([1, -1, 1, 3950672395, 3782658992397]\) | \(598672364899527954087397631/346996861747253448998400\) | \(-3952511128339808817497400000000\) | \([2]\) | \(127401984\) | \(4.5604\) |
Rank
sage: E.rank()
The elliptic curves in class 72450.cy have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.cy do not have complex multiplication.Modular form 72450.2.a.cy
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 3 & 6 & 12 & 12 \\ 2 & 1 & 2 & 2 & 6 & 3 & 6 & 6 \\ 4 & 2 & 1 & 4 & 12 & 6 & 3 & 12 \\ 4 & 2 & 4 & 1 & 12 & 6 & 12 & 3 \\ 3 & 6 & 12 & 12 & 1 & 2 & 4 & 4 \\ 6 & 3 & 6 & 6 & 2 & 1 & 2 & 2 \\ 12 & 6 & 3 & 12 & 4 & 2 & 1 & 4 \\ 12 & 6 & 12 & 3 & 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
