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SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 221760.dp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221760.dp1 | 221760ks7 | \([0, 0, 0, -3066624588, 65364062085488]\) | \(16689299266861680229173649/2396798250\) | \(458035278446592000\) | \([2]\) | \(63700992\) | \(3.7114\) | |
| 221760.dp2 | 221760ks8 | \([0, 0, 0, -196704588, 964760517488]\) | \(4404531606962679693649/444872222400201750\) | \(85016405640412417425408000\) | \([2]\) | \(63700992\) | \(3.7114\) | |
| 221760.dp3 | 221760ks6 | \([0, 0, 0, -191664588, 1021307301488]\) | \(4074571110566294433649/48828650062500\) | \(9331300341006336000000\) | \([2, 2]\) | \(31850496\) | \(3.3649\) | |
| 221760.dp4 | 221760ks5 | \([0, 0, 0, -43206348, -109096186768]\) | \(46676570542430835889/106752955783320\) | \(20400807546988863160320\) | \([2]\) | \(21233664\) | \(3.1621\) | |
| 221760.dp5 | 221760ks4 | \([0, 0, 0, -37907148, 89425945712]\) | \(31522423139920199089/164434491947880\) | \(31423920768287904890880\) | \([2]\) | \(21233664\) | \(3.1621\) | |
| 221760.dp6 | 221760ks3 | \([0, 0, 0, -11664588, 16835301488]\) | \(-918468938249433649/109183593750000\) | \(-20865309696000000000000\) | \([2]\) | \(15925248\) | \(3.0183\) | |
| 221760.dp7 | 221760ks2 | \([0, 0, 0, -3692748, -338954128]\) | \(29141055407581489/16604321025600\) | \(3173135162451532185600\) | \([2, 2]\) | \(10616832\) | \(2.8156\) | |
| 221760.dp8 | 221760ks1 | \([0, 0, 0, 915252, -42198928]\) | \(443688652450511/260789760000\) | \(-49837699246325760000\) | \([2]\) | \(5308416\) | \(2.4690\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 221760.dp have rank \(1\).
Complex multiplication
The elliptic curves in class 221760.dp do not have complex multiplication.Modular form 221760.2.a.dp
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.
