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SageMath
E = EllipticCurve("dy1")
E.isogeny_class()
Elliptic curves in class 73920.dy
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 73920.dy1 | 73920bs7 | \([0, -1, 0, -340736065, -2420777609663]\) | \(16689299266861680229173649/2396798250\) | \(628306280448000\) | \([2]\) | \(7962624\) | \(3.1621\) | |
| 73920.dy2 | 73920bs8 | \([0, -1, 0, -21856065, -35724585663]\) | \(4404531606962679693649/444872222400201750\) | \(116620583868878487552000\) | \([2]\) | \(7962624\) | \(3.1621\) | |
| 73920.dy3 | 73920bs6 | \([0, -1, 0, -21296065, -37819097663]\) | \(4074571110566294433649/48828650062500\) | \(12800137641984000000\) | \([2, 2]\) | \(3981312\) | \(2.8156\) | |
| 73920.dy4 | 73920bs5 | \([0, -1, 0, -4800705, 4042199745]\) | \(46676570542430835889/106752955783320\) | \(27984646840862638080\) | \([2]\) | \(2654208\) | \(2.6128\) | |
| 73920.dy5 | 73920bs4 | \([0, -1, 0, -4211905, -3310668095]\) | \(31522423139920199089/164434491947880\) | \(43105515457185054720\) | \([2]\) | \(2654208\) | \(2.6128\) | |
| 73920.dy6 | 73920bs3 | \([0, -1, 0, -1296065, -623097663]\) | \(-918468938249433649/109183593750000\) | \(-28621824000000000000\) | \([2]\) | \(1990656\) | \(2.4690\) | |
| 73920.dy7 | 73920bs2 | \([0, -1, 0, -410305, 12690625]\) | \(29141055407581489/16604321025600\) | \(4352723130934886400\) | \([2, 2]\) | \(1327104\) | \(2.2663\) | |
| 73920.dy8 | 73920bs1 | \([0, -1, 0, 101695, 1529025]\) | \(443688652450511/260789760000\) | \(-68364470845440000\) | \([2]\) | \(663552\) | \(1.9197\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 73920.dy have rank \(0\).
Complex multiplication
The elliptic curves in class 73920.dy do not have complex multiplication.Modular form 73920.2.a.dy
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 4 & 6 & 12 \\ 4 & 1 & 2 & 3 & 12 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 12 & 3 & 6 & 1 & 4 & 12 & 2 & 4 \\ 3 & 12 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
