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SageMath
E = EllipticCurve("ir1")
E.isogeny_class()
Elliptic curves in class 277200ir
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 277200.ir8 | 277200ir1 | \([0, 0, 0, 5720325, 659358250]\) | \(443688652450511/260789760000\) | \(-12167407042560000000000\) | \([2]\) | \(15925248\) | \(2.9271\) | \(\Gamma_0(N)\)-optimal |
| 277200.ir7 | 277200ir2 | \([0, 0, 0, -23079675, 5296158250]\) | \(29141055407581489/16604321025600\) | \(774691201770393600000000\) | \([2, 2]\) | \(31850496\) | \(3.2737\) | |
| 277200.ir6 | 277200ir3 | \([0, 0, 0, -72903675, -263051585750]\) | \(-918468938249433649/109183593750000\) | \(-5094069750000000000000000\) | \([2]\) | \(47775744\) | \(3.4764\) | |
| 277200.ir4 | 277200ir4 | \([0, 0, 0, -270039675, 1704627918250]\) | \(46676570542430835889/106752955783320\) | \(4980665905026577920000000\) | \([2]\) | \(63700992\) | \(3.6203\) | |
| 277200.ir5 | 277200ir5 | \([0, 0, 0, -236919675, -1397280401750]\) | \(31522423139920199089/164434491947880\) | \(7671855656320289280000000\) | \([2]\) | \(63700992\) | \(3.6203\) | |
| 277200.ir3 | 277200ir6 | \([0, 0, 0, -1197903675, -15957926585750]\) | \(4074571110566294433649/48828650062500\) | \(2278149497316000000000000\) | \([2, 2]\) | \(95551488\) | \(3.8230\) | |
| 277200.ir2 | 277200ir7 | \([0, 0, 0, -1229403675, -15074383085750]\) | \(4404531606962679693649/444872222400201750\) | \(20755958408303812848000000000\) | \([2]\) | \(191102976\) | \(4.1696\) | |
| 277200.ir1 | 277200ir8 | \([0, 0, 0, -19166403675, -1021313470085750]\) | \(16689299266861680229173649/2396798250\) | \(111825019152000000000\) | \([2]\) | \(191102976\) | \(4.1696\) |
Rank
sage: E.rank()
The elliptic curves in class 277200ir have rank \(0\).
Complex multiplication
The elliptic curves in class 277200ir do not have complex multiplication.Modular form 277200.2.a.ir
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.
