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SageMath
E = EllipticCurve("ig1")
E.isogeny_class()
Elliptic curves in class 270480.ig
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
270480.ig1 | 270480ig7 | \([0, 1, 0, -199056828560, -32072636951597292]\) | \(1810117493172631097464564372609/125368453502655029296875000\) | \(60413842170404296875000000000000000\) | \([2]\) | \(2293235712\) | \(5.4218\) | |
270480.ig2 | 270480ig6 | \([0, 1, 0, -195622814480, -33302452172433900]\) | \(1718043013877225552292911401729/9180538178765625000000\) | \(4424012333848973376000000000000\) | \([2, 2]\) | \(1146617856\) | \(5.0753\) | |
270480.ig3 | 270480ig3 | \([0, 1, 0, -195622563600, -33302541862134252]\) | \(1718036403880129446396978632449/49057344000000\) | \(23640258413592576000000\) | \([2]\) | \(573308928\) | \(4.7287\) | |
270480.ig4 | 270480ig8 | \([0, 1, 0, -192192814480, -34526527248433900]\) | \(-1629247127728109256861881401729/125809119536174660320875000\) | \(-60626194859259546059123191296000000\) | \([4]\) | \(2293235712\) | \(5.4218\) | |
270480.ig5 | 270480ig4 | \([0, 1, 0, -37096509200, 2740652917972500]\) | \(11715873038622856702991202049/46415372499833400000000\) | \(22367117964217957075353600000000\) | \([2]\) | \(764411904\) | \(4.8725\) | |
270480.ig6 | 270480ig2 | \([0, 1, 0, -3444469520, -3078642033132]\) | \(9378698233516887309850369/5418996968417034240000\) | \(2611362096485563028692008960000\) | \([2, 2]\) | \(382205952\) | \(4.5260\) | |
270480.ig7 | 270480ig1 | \([0, 1, 0, -2416865040, -45612835627500]\) | \(3239908336204082689644289/9880281924658790400\) | \(4761212060279529602128281600\) | \([2]\) | \(191102976\) | \(4.1794\) | \(\Gamma_0(N)\)-optimal |
270480.ig8 | 270480ig5 | \([0, 1, 0, 13765898480, -24605370327532]\) | \(598672364899527954087397631/346996861747253448998400\) | \(-167214423194429935702887471513600\) | \([4]\) | \(764411904\) | \(4.8725\) |
Rank
sage: E.rank()
The elliptic curves in class 270480.ig have rank \(0\).
Complex multiplication
The elliptic curves in class 270480.ig do not have complex multiplication.Modular form 270480.2.a.ig
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.