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Properties

Label 22.10.210...000.1
Degree $22$
Signature $[10, 6]$
Discriminant $2.108\times 10^{194}$
Root discriminant \(6.806\times 10^{8}\)
Ramified primes $2,3,5,113,337,310501,11155561$
Class number not computed
Class group not computed
Galois group $C_2^{10}.M_{11}$ (as 22T43)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 1521110995119*x^20 + 20767967363363151457665*x^18 - 125802676272554404269617160982350*x^16 - 447129861314250111523595651626294337559000*x^14 + 2211187005788657175999069028194179567572695016920000*x^12 + 495161089736201224478016513692993953268704949758149081450000*x^10 - 6532269446805556223473454007587119713902772265719196261716243847500000*x^8 + 880101110859184651857903730474356017770939048207067136486977359706942040000000*x^6 + 5695692754570027594526557200758139973678062841883039454997024866872864512936560150000000*x^4 - 593268499597857685637766137050940784311360565636847362570893397339081071862088416668321250000000*x^2 - 1495722903681177175207908834306985386050749026948136139791810294784018587329257867815333630665502500000000)
 
gp: K = bnfinit(y^22 + 1521110995119*y^20 + 20767967363363151457665*y^18 - 125802676272554404269617160982350*y^16 - 447129861314250111523595651626294337559000*y^14 + 2211187005788657175999069028194179567572695016920000*y^12 + 495161089736201224478016513692993953268704949758149081450000*y^10 - 6532269446805556223473454007587119713902772265719196261716243847500000*y^8 + 880101110859184651857903730474356017770939048207067136486977359706942040000000*y^6 + 5695692754570027594526557200758139973678062841883039454997024866872864512936560150000000*y^4 - 593268499597857685637766137050940784311360565636847362570893397339081071862088416668321250000000*y^2 - 1495722903681177175207908834306985386050749026948136139791810294784018587329257867815333630665502500000000, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 1521110995119*x^20 + 20767967363363151457665*x^18 - 125802676272554404269617160982350*x^16 - 447129861314250111523595651626294337559000*x^14 + 2211187005788657175999069028194179567572695016920000*x^12 + 495161089736201224478016513692993953268704949758149081450000*x^10 - 6532269446805556223473454007587119713902772265719196261716243847500000*x^8 + 880101110859184651857903730474356017770939048207067136486977359706942040000000*x^6 + 5695692754570027594526557200758139973678062841883039454997024866872864512936560150000000*x^4 - 593268499597857685637766137050940784311360565636847362570893397339081071862088416668321250000000*x^2 - 1495722903681177175207908834306985386050749026948136139791810294784018587329257867815333630665502500000000);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 1521110995119*x^20 + 20767967363363151457665*x^18 - 125802676272554404269617160982350*x^16 - 447129861314250111523595651626294337559000*x^14 + 2211187005788657175999069028194179567572695016920000*x^12 + 495161089736201224478016513692993953268704949758149081450000*x^10 - 6532269446805556223473454007587119713902772265719196261716243847500000*x^8 + 880101110859184651857903730474356017770939048207067136486977359706942040000000*x^6 + 5695692754570027594526557200758139973678062841883039454997024866872864512936560150000000*x^4 - 593268499597857685637766137050940784311360565636847362570893397339081071862088416668321250000000*x^2 - 1495722903681177175207908834306985386050749026948136139791810294784018587329257867815333630665502500000000)
 

\( x^{22} + 1521110995119 x^{20} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $22$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[10, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(210\!\cdots\!000\) \(\medspace = 2^{72}\cdot 3^{22}\cdot 5^{10}\cdot 113^{10}\cdot 337^{8}\cdot 310501^{8}\cdot 11155561^{10}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(6.806\times 10^{8}\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(3\), \(5\), \(113\), \(337\), \(310501\), \(11155561\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{18908675895}a^{4}+\frac{2805641173}{6302891965}a^{2}$, $\frac{1}{56726027685}a^{5}+\frac{2805641173}{18908675895}a^{3}$, $\frac{1}{35\!\cdots\!25}a^{6}+\frac{2805641173}{11\!\cdots\!75}a^{4}+\frac{2995107039}{6302891965}a^{2}$, $\frac{1}{21\!\cdots\!50}a^{7}+\frac{2805641173}{71\!\cdots\!50}a^{5}+\frac{998369013}{12605783930}a^{3}$, $\frac{1}{13\!\cdots\!50}a^{8}+\frac{2805641173}{45\!\cdots\!50}a^{6}+\frac{998369013}{79\!\cdots\!50}a^{4}-\frac{1313928607}{6302891965}a^{2}$, $\frac{1}{81\!\cdots\!00}a^{9}+\frac{2805641173}{27\!\cdots\!00}a^{7}-\frac{9610676891}{14\!\cdots\!00}a^{5}+\frac{436664437}{7563470358}a^{3}$, $\frac{1}{51\!\cdots\!00}a^{10}+\frac{2805641173}{17\!\cdots\!00}a^{8}-\frac{9610676891}{90\!\cdots\!00}a^{6}+\frac{436664437}{47\!\cdots\!70}a^{4}+\frac{1136207753}{6302891965}a^{2}$, $\frac{1}{10\!\cdots\!00}a^{11}-\frac{4188860411}{10\!\cdots\!00}a^{9}-\frac{9231745159}{54\!\cdots\!00}a^{7}+\frac{4583070931}{71\!\cdots\!50}a^{5}+\frac{47158898}{1260578393}a^{3}$, $\frac{1}{19\!\cdots\!00}a^{12}+\frac{2805641173}{64\!\cdots\!00}a^{10}-\frac{9610676891}{34\!\cdots\!00}a^{8}-\frac{231610261}{20\!\cdots\!00}a^{6}-\frac{166943342}{23\!\cdots\!35}a^{4}-\frac{1991861411}{6302891965}a^{2}$, $\frac{1}{19\!\cdots\!00}a^{13}+\frac{2805641173}{64\!\cdots\!00}a^{11}-\frac{3620462813}{10\!\cdots\!00}a^{9}-\frac{49392781}{30\!\cdots\!00}a^{7}+\frac{143923565}{28\!\cdots\!82}a^{5}+\frac{669495388}{6302891965}a^{3}$, $\frac{1}{73\!\cdots\!00}a^{14}+\frac{2805641173}{24\!\cdots\!00}a^{12}-\frac{9610676891}{12\!\cdots\!00}a^{10}-\frac{231610261}{75\!\cdots\!00}a^{8}-\frac{83471671}{45\!\cdots\!25}a^{6}+\frac{2155515277}{11\!\cdots\!75}a^{4}+\frac{814727692}{6302891965}a^{2}$, $\frac{1}{73\!\cdots\!00}a^{15}+\frac{2805641173}{24\!\cdots\!00}a^{13}+\frac{332789671}{14\!\cdots\!00}a^{11}+\frac{295138268}{12\!\cdots\!25}a^{9}-\frac{2414498641}{54\!\cdots\!00}a^{7}+\frac{6200294513}{14\!\cdots\!00}a^{5}+\frac{247315717}{37817351790}a^{3}$, $\frac{1}{27\!\cdots\!00}a^{16}+\frac{2805641173}{92\!\cdots\!00}a^{14}-\frac{9610676891}{48\!\cdots\!00}a^{12}-\frac{231610261}{28\!\cdots\!00}a^{10}-\frac{83471671}{17\!\cdots\!50}a^{8}+\frac{2155515277}{45\!\cdots\!50}a^{6}-\frac{1829388091}{79\!\cdots\!50}a^{4}+\frac{2485191236}{6302891965}a^{2}$, $\frac{1}{27\!\cdots\!00}a^{17}+\frac{2805641173}{92\!\cdots\!00}a^{15}-\frac{9610676891}{48\!\cdots\!00}a^{13}+\frac{436664437}{25\!\cdots\!00}a^{11}+\frac{1136207753}{34\!\cdots\!00}a^{9}-\frac{988615783}{18\!\cdots\!00}a^{7}+\frac{3812777569}{47\!\cdots\!00}a^{5}+\frac{762563167}{18908675895}a^{3}$, $\frac{1}{10\!\cdots\!00}a^{18}+\frac{2805641173}{34\!\cdots\!00}a^{16}-\frac{9610676891}{18\!\cdots\!00}a^{14}-\frac{231610261}{10\!\cdots\!00}a^{12}-\frac{83471671}{64\!\cdots\!00}a^{10}+\frac{2155515277}{17\!\cdots\!00}a^{8}+\frac{7117619657}{90\!\cdots\!00}a^{6}+\frac{1763610803}{79\!\cdots\!50}a^{4}+\frac{1823931483}{6302891965}a^{2}$, $\frac{1}{10\!\cdots\!00}a^{19}+\frac{2805641173}{34\!\cdots\!00}a^{17}-\frac{9610676891}{18\!\cdots\!00}a^{15}-\frac{231610261}{10\!\cdots\!00}a^{13}-\frac{83471671}{64\!\cdots\!00}a^{11}+\frac{81826933}{25\!\cdots\!50}a^{9}+\frac{495119489}{22\!\cdots\!25}a^{7}+\frac{10532821139}{14\!\cdots\!00}a^{5}+\frac{76938664}{18908675895}a^{3}$, $\frac{1}{57\!\cdots\!00}a^{20}-\frac{28\!\cdots\!19}{63\!\cdots\!00}a^{18}+\frac{52\!\cdots\!01}{33\!\cdots\!00}a^{16}-\frac{11\!\cdots\!29}{26\!\cdots\!00}a^{14}-\frac{10\!\cdots\!73}{78\!\cdots\!00}a^{12}+\frac{33\!\cdots\!43}{12\!\cdots\!00}a^{10}+\frac{12\!\cdots\!73}{49\!\cdots\!00}a^{8}-\frac{12\!\cdots\!37}{34\!\cdots\!00}a^{6}-\frac{14\!\cdots\!97}{57\!\cdots\!25}a^{4}+\frac{78\!\cdots\!76}{27\!\cdots\!55}a^{2}+\frac{29\!\cdots\!80}{14\!\cdots\!29}$, $\frac{1}{57\!\cdots\!00}a^{21}-\frac{28\!\cdots\!19}{63\!\cdots\!00}a^{19}+\frac{52\!\cdots\!01}{33\!\cdots\!00}a^{17}-\frac{11\!\cdots\!29}{26\!\cdots\!00}a^{15}-\frac{10\!\cdots\!73}{78\!\cdots\!00}a^{13}+\frac{33\!\cdots\!43}{12\!\cdots\!00}a^{11}+\frac{12\!\cdots\!79}{14\!\cdots\!00}a^{9}-\frac{12\!\cdots\!51}{11\!\cdots\!00}a^{7}-\frac{81\!\cdots\!81}{11\!\cdots\!05}a^{5}-\frac{44\!\cdots\!17}{54\!\cdots\!10}a^{3}+\frac{29\!\cdots\!80}{14\!\cdots\!29}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 + 1521110995119*x^20 + 20767967363363151457665*x^18 - 125802676272554404269617160982350*x^16 - 447129861314250111523595651626294337559000*x^14 + 2211187005788657175999069028194179567572695016920000*x^12 + 495161089736201224478016513692993953268704949758149081450000*x^10 - 6532269446805556223473454007587119713902772265719196261716243847500000*x^8 + 880101110859184651857903730474356017770939048207067136486977359706942040000000*x^6 + 5695692754570027594526557200758139973678062841883039454997024866872864512936560150000000*x^4 - 593268499597857685637766137050940784311360565636847362570893397339081071862088416668321250000000*x^2 - 1495722903681177175207908834306985386050749026948136139791810294784018587329257867815333630665502500000000)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^22 + 1521110995119*x^20 + 20767967363363151457665*x^18 - 125802676272554404269617160982350*x^16 - 447129861314250111523595651626294337559000*x^14 + 2211187005788657175999069028194179567572695016920000*x^12 + 495161089736201224478016513692993953268704949758149081450000*x^10 - 6532269446805556223473454007587119713902772265719196261716243847500000*x^8 + 880101110859184651857903730474356017770939048207067136486977359706942040000000*x^6 + 5695692754570027594526557200758139973678062841883039454997024866872864512936560150000000*x^4 - 593268499597857685637766137050940784311360565636847362570893397339081071862088416668321250000000*x^2 - 1495722903681177175207908834306985386050749026948136139791810294784018587329257867815333630665502500000000, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^22 + 1521110995119*x^20 + 20767967363363151457665*x^18 - 125802676272554404269617160982350*x^16 - 447129861314250111523595651626294337559000*x^14 + 2211187005788657175999069028194179567572695016920000*x^12 + 495161089736201224478016513692993953268704949758149081450000*x^10 - 6532269446805556223473454007587119713902772265719196261716243847500000*x^8 + 880101110859184651857903730474356017770939048207067136486977359706942040000000*x^6 + 5695692754570027594526557200758139973678062841883039454997024866872864512936560150000000*x^4 - 593268499597857685637766137050940784311360565636847362570893397339081071862088416668321250000000*x^2 - 1495722903681177175207908834306985386050749026948136139791810294784018587329257867815333630665502500000000);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 1521110995119*x^20 + 20767967363363151457665*x^18 - 125802676272554404269617160982350*x^16 - 447129861314250111523595651626294337559000*x^14 + 2211187005788657175999069028194179567572695016920000*x^12 + 495161089736201224478016513692993953268704949758149081450000*x^10 - 6532269446805556223473454007587119713902772265719196261716243847500000*x^8 + 880101110859184651857903730474356017770939048207067136486977359706942040000000*x^6 + 5695692754570027594526557200758139973678062841883039454997024866872864512936560150000000*x^4 - 593268499597857685637766137050940784311360565636847362570893397339081071862088416668321250000000*x^2 - 1495722903681177175207908834306985386050749026948136139791810294784018587329257867815333630665502500000000);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^{10}.M_{11}$ (as 22T43):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 8110080
The 52 conjugacy class representatives for $C_2^{10}.M_{11}$
Character table for $C_2^{10}.M_{11}$

Intermediate fields

11.11.118769262421915560193703211428553337469927424.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 22 sibling: data not computed
Degree 44 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.11.0.1}{11} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ ${\href{/padicField/13.11.0.1}{11} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ $16{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ $16{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.11.0.1}{11} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.3.2$x^{2} + 4 x + 10$$2$$1$$3$$C_2$$[3]$
2.4.4.4$x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
Deg $16$$16$$1$$65$
\(3\) Copy content Toggle raw display 3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $18$$9$$2$$20$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.16.8.1$x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$
\(113\) Copy content Toggle raw display $\Q_{113}$$x + 110$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 110$$1$$1$$0$Trivial$[\ ]$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.8.4.1$x^{8} + 37516 x^{7} + 527794298 x^{6} + 3300131705804 x^{5} + 7738073280934187 x^{4} + 372947606044816 x^{3} + 211347971757704 x^{2} + 479763102267371150 x + 28993879005127045$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
113.8.4.1$x^{8} + 37516 x^{7} + 527794298 x^{6} + 3300131705804 x^{5} + 7738073280934187 x^{4} + 372947606044816 x^{3} + 211347971757704 x^{2} + 479763102267371150 x + 28993879005127045$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(337\) Copy content Toggle raw display $\Q_{337}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{337}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $16$$2$$8$$8$
\(310501\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $6$$2$$3$$3$
Deg $6$$2$$3$$3$
\(11155561\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $8$$2$$4$$4$
Deg $8$$2$$4$$4$