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SageMath
E = EllipticCurve("cv1")
E.isogeny_class()
Elliptic curves in class 68970.cv
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68970.cv1 | 68970cv8 | \([1, 0, 0, -2592338185, -50459185341223]\) | \(1087533321226184807035053481/8484255812957933638080\) | \(15030376712259569873810642880\) | \([2]\) | \(92897280\) | \(4.2352\) | |
| 68970.cv2 | 68970cv5 | \([1, 0, 0, -2587464910, -50659589377528]\) | \(1081411559614045490773061881/522522049500\) | \(925679684534269500\) | \([2]\) | \(30965760\) | \(3.6859\) | |
| 68970.cv3 | 68970cv6 | \([1, 0, 0, -273203785, 431436105497]\) | \(1272998045160051207059881/691293848290254950400\) | \(1224669221170932350185574400\) | \([2, 2]\) | \(46448640\) | \(3.8886\) | |
| 68970.cv4 | 68970cv3 | \([1, 0, 0, -211251785, 1180299491097]\) | \(588530213343917460371881/861551575695360000\) | \(1526291170990447656960000\) | \([4]\) | \(23224320\) | \(3.5420\) | |
| 68970.cv5 | 68970cv2 | \([1, 0, 0, -161717410, -791557422028]\) | \(264020672568758737421881/5803468580250000\) | \(10281198601496270250000\) | \([2, 2]\) | \(15482880\) | \(3.3393\) | |
| 68970.cv6 | 68970cv4 | \([1, 0, 0, -155969910, -850424466528]\) | \(-236859095231405581781881/39282983014374049500\) | \(-69592200671927505506269500\) | \([2]\) | \(30965760\) | \(3.6859\) | |
| 68970.cv7 | 68970cv1 | \([1, 0, 0, -10467410, -11440172028]\) | \(71595431380957421881/9522562500000000\) | \(16869800345062500000000\) | \([4]\) | \(7741440\) | \(2.9927\) | \(\Gamma_0(N)\)-optimal |
| 68970.cv8 | 68970cv7 | \([1, 0, 0, 1054698615, 3395048681817]\) | \(73240740785321709623685719/45195275784938365817280\) | \(-80066187964841196285626374080\) | \([2]\) | \(92897280\) | \(4.2352\) |
Rank
sage: E.rank()
The elliptic curves in class 68970.cv have rank \(1\).
Complex multiplication
The elliptic curves in class 68970.cv do not have complex multiplication.Modular form 68970.2.a.cv
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
