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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 248430cn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 248430.cn7 | 248430cn1 | \([1, 1, 0, 1738838, -664302764]\) | \(1023887723039/928972800\) | \(-527535089102433484800\) | \([2]\) | \(14155776\) | \(2.6636\) | \(\Gamma_0(N)\)-optimal |
| 248430.cn6 | 248430cn2 | \([1, 1, 0, -8860842, -5957782956]\) | \(135487869158881/51438240000\) | \(29210194875105447840000\) | \([2, 2]\) | \(28311552\) | \(3.0101\) | |
| 248430.cn5 | 248430cn3 | \([1, 1, 0, -62521722, 186008649156]\) | \(47595748626367201/1215506250000\) | \(690248625038660756250000\) | \([2, 2]\) | \(56623104\) | \(3.3567\) | |
| 248430.cn4 | 248430cn4 | \([1, 1, 0, -124794842, -536494953756]\) | \(378499465220294881/120530818800\) | \(68445745919845301170800\) | \([2]\) | \(56623104\) | \(3.3567\) | |
| 248430.cn2 | 248430cn5 | \([1, 1, 0, -994134222, 12064254346656]\) | \(191342053882402567201/129708022500\) | \(73657197720792198922500\) | \([2, 2]\) | \(113246208\) | \(3.7033\) | |
| 248430.cn8 | 248430cn6 | \([1, 1, 0, 10516698, 594687824424]\) | \(226523624554079/269165039062500\) | \(-152850549408008422851562500\) | \([2]\) | \(113246208\) | \(3.7033\) | |
| 248430.cn1 | 248430cn7 | \([1, 1, 0, -15906144972, 772132424676306]\) | \(783736670177727068275201/360150\) | \(204518111122566150\) | \([2]\) | \(226492416\) | \(4.0499\) | |
| 248430.cn3 | 248430cn8 | \([1, 1, 0, -987923472, 12222440907006]\) | \(-187778242790732059201/4984939585440150\) | \(-2830793913853470594270846150\) | \([2]\) | \(226492416\) | \(4.0499\) |
Rank
sage: E.rank()
The elliptic curves in class 248430cn have rank \(1\).
Complex multiplication
The elliptic curves in class 248430cn do not have complex multiplication.Modular form 248430.2.a.cn
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
