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SageMath
E = EllipticCurve("pw1")
E.isogeny_class()
Elliptic curves in class 369600pw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 369600.pw8 | 369600pw1 | \([0, 1, 0, 2542367, 196212863]\) | \(443688652450511/260789760000\) | \(-1068194856960000000000\) | \([2]\) | \(15925248\) | \(2.7244\) | \(\Gamma_0(N)\)-optimal |
| 369600.pw7 | 369600pw2 | \([0, 1, 0, -10257633, 1565812863]\) | \(29141055407581489/16604321025600\) | \(68011298920857600000000\) | \([2, 2]\) | \(31850496\) | \(3.0710\) | |
| 369600.pw6 | 369600pw3 | \([0, 1, 0, -32401633, -77952011137]\) | \(-918468938249433649/109183593750000\) | \(-447216000000000000000000\) | \([2]\) | \(47775744\) | \(3.2737\) | |
| 369600.pw4 | 369600pw4 | \([0, 1, 0, -120017633, 505034932863]\) | \(46676570542430835889/106752955783320\) | \(437260106888478720000000\) | \([4]\) | \(63700992\) | \(3.4176\) | |
| 369600.pw5 | 369600pw5 | \([0, 1, 0, -105297633, -414044107137]\) | \(31522423139920199089/164434491947880\) | \(673523679018516480000000\) | \([2]\) | \(63700992\) | \(3.4176\) | |
| 369600.pw3 | 369600pw6 | \([0, 1, 0, -532401633, -4728452011137]\) | \(4074571110566294433649/48828650062500\) | \(200002150656000000000000\) | \([2, 2]\) | \(95551488\) | \(3.6203\) | |
| 369600.pw2 | 369600pw7 | \([0, 1, 0, -546401633, -4466666011137]\) | \(4404531606962679693649/444872222400201750\) | \(1822196622951226368000000000\) | \([4]\) | \(191102976\) | \(3.9669\) | |
| 369600.pw1 | 369600pw8 | \([0, 1, 0, -8518401633, -302614238011137]\) | \(16689299266861680229173649/2396798250\) | \(9817285632000000000\) | \([2]\) | \(191102976\) | \(3.9669\) |
Rank
sage: E.rank()
The elliptic curves in class 369600pw have rank \(1\).
Complex multiplication
The elliptic curves in class 369600pw do not have complex multiplication.Modular form 369600.2.a.pw
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
