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SageMath
E = EllipticCurve("hq1")
E.isogeny_class()
Elliptic curves in class 177870hq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177870.bu8 | 177870hq1 | \([1, 1, 0, 9421058, 1397530996]\) | \(443688652450511/260789760000\) | \(-54354422482039088640000\) | \([2]\) | \(19906560\) | \(3.0519\) | \(\Gamma_0(N)\)-optimal |
| 177870.bu7 | 177870hq2 | \([1, 1, 0, -38010942, 11178009396]\) | \(29141055407581489/16604321025600\) | \(3460712107917377499278400\) | \([2, 2]\) | \(39813120\) | \(3.3984\) | |
| 177870.bu6 | 177870hq3 | \([1, 1, 0, -120068302, -556030258076]\) | \(-918468938249433649/109183593750000\) | \(-22756304476045464843750000\) | \([2]\) | \(59719680\) | \(3.6012\) | |
| 177870.bu4 | 177870hq4 | \([1, 1, 0, -444740342, 3602679957276]\) | \(46676570542430835889/106752955783320\) | \(22249705125895331766315480\) | \([2]\) | \(79626240\) | \(3.7450\) | |
| 177870.bu5 | 177870hq5 | \([1, 1, 0, -390193542, -2953424680884]\) | \(31522423139920199089/164434491947880\) | \(34271828180502655337761320\) | \([2]\) | \(79626240\) | \(3.7450\) | |
| 177870.bu3 | 177870hq6 | \([1, 1, 0, -1972880802, -33729155820576]\) | \(4074571110566294433649/48828650062500\) | \(10176983462559148605562500\) | \([2, 2]\) | \(119439360\) | \(3.9478\) | |
| 177870.bu2 | 177870hq7 | \([1, 1, 0, -2024759552, -31861738711326]\) | \(4404531606962679693649/444872222400201750\) | \(92721327428132989008782955750\) | \([2]\) | \(238878720\) | \(4.2943\) | |
| 177870.bu1 | 177870hq8 | \([1, 1, 0, -31566002052, -2158639552679826]\) | \(16689299266861680229173649/2396798250\) | \(499546395858150044250\) | \([2]\) | \(238878720\) | \(4.2943\) |
Rank
sage: E.rank()
The elliptic curves in class 177870hq have rank \(0\).
Complex multiplication
The elliptic curves in class 177870hq do not have complex multiplication.Modular form 177870.2.a.hq
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.
