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Properties

Label 16.10.298...875.2
Degree $16$
Signature $[10, 3]$
Discriminant $-2.989\times 10^{20}$
Root discriminant \(19.04\)
Ramified primes $3,5,179,1021$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_4^4.C_2\wr C_4$ (as 16T1771)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 11*x^14 + 11*x^13 - 75*x^12 + 65*x^11 + 125*x^10 - 187*x^9 - 68*x^8 + 128*x^7 + 63*x^6 - 20*x^5 - 88*x^4 + 33*x^3 + 16*x^2 - 9*x + 1)
 
gp: K = bnfinit(y^16 - 6*y^15 + 11*y^14 + 11*y^13 - 75*y^12 + 65*y^11 + 125*y^10 - 187*y^9 - 68*y^8 + 128*y^7 + 63*y^6 - 20*y^5 - 88*y^4 + 33*y^3 + 16*y^2 - 9*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 11*x^14 + 11*x^13 - 75*x^12 + 65*x^11 + 125*x^10 - 187*x^9 - 68*x^8 + 128*x^7 + 63*x^6 - 20*x^5 - 88*x^4 + 33*x^3 + 16*x^2 - 9*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 11*x^14 + 11*x^13 - 75*x^12 + 65*x^11 + 125*x^10 - 187*x^9 - 68*x^8 + 128*x^7 + 63*x^6 - 20*x^5 - 88*x^4 + 33*x^3 + 16*x^2 - 9*x + 1)
 

\( x^{16} - 6 x^{15} + 11 x^{14} + 11 x^{13} - 75 x^{12} + 65 x^{11} + 125 x^{10} - 187 x^{9} - 68 x^{8} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[10, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-298892216010498046875\) \(\medspace = -\,3^{8}\cdot 5^{12}\cdot 179\cdot 1021^{2}\) 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:  \(19.04\)
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:  $3^{1/2}5^{3/4}179^{1/2}1021^{1/2}\approx 2475.868173489506$
Ramified primes:   \(3\), \(5\), \(179\), \(1021\) 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:  \(\Q(\sqrt{-179}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1087738199}a^{15}-\frac{320415255}{1087738199}a^{14}-\frac{159202311}{1087738199}a^{13}+\frac{166895705}{1087738199}a^{12}+\frac{333731880}{1087738199}a^{11}+\frac{410179281}{1087738199}a^{10}-\frac{4712343}{35088329}a^{9}+\frac{108302219}{1087738199}a^{8}-\frac{362029204}{1087738199}a^{7}-\frac{263883464}{1087738199}a^{6}+\frac{416758401}{1087738199}a^{5}+\frac{251174929}{1087738199}a^{4}+\frac{216134344}{1087738199}a^{3}+\frac{14898642}{1087738199}a^{2}+\frac{6348602}{35088329}a-\frac{9399023}{1087738199}$ 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

Trivial group, which has order $1$ (assuming GRH)

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:  $12$
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:   $\frac{38675649}{35088329}a^{15}-\frac{220162298}{35088329}a^{14}+\frac{359599288}{35088329}a^{13}+\frac{528774725}{35088329}a^{12}-\frac{2737579388}{35088329}a^{11}+\frac{1717606926}{35088329}a^{10}+\frac{5288324002}{35088329}a^{9}-\frac{5718955480}{35088329}a^{8}-\frac{4107617906}{35088329}a^{7}+\frac{3845459058}{35088329}a^{6}+\frac{3242604861}{35088329}a^{5}+\frac{31672381}{35088329}a^{4}-\frac{3259758314}{35088329}a^{3}+\frac{460718207}{35088329}a^{2}+\frac{768399610}{35088329}a-\frac{190997351}{35088329}$, $\frac{2315963747}{1087738199}a^{15}-\frac{13050805569}{1087738199}a^{14}+\frac{20814808591}{1087738199}a^{13}+\frac{32477228510}{1087738199}a^{12}-\frac{160887962231}{1087738199}a^{11}+\frac{93377821685}{1087738199}a^{10}+\frac{10191156420}{35088329}a^{9}-\frac{313203901056}{1087738199}a^{8}-\frac{256214574016}{1087738199}a^{7}+\frac{187736038496}{1087738199}a^{6}+\frac{201499358195}{1087738199}a^{5}+\frac{33144852209}{1087738199}a^{4}-\frac{181757967242}{1087738199}a^{3}+\frac{15506832885}{1087738199}a^{2}+\frac{1153020144}{35088329}a-\frac{6880286156}{1087738199}$, $\frac{2449622833}{1087738199}a^{15}-\frac{13295083882}{1087738199}a^{14}+\frac{19358132186}{1087738199}a^{13}+\frac{38076449144}{1087738199}a^{12}-\frac{162662843246}{1087738199}a^{11}+\frac{67834274881}{1087738199}a^{10}+\frac{11150912994}{35088329}a^{9}-\frac{269446087103}{1087738199}a^{8}-\frac{313568611129}{1087738199}a^{7}+\frac{149395976157}{1087738199}a^{6}+\frac{227140911693}{1087738199}a^{5}+\frac{67933021142}{1087738199}a^{4}-\frac{174960013110}{1087738199}a^{3}-\frac{9275766765}{1087738199}a^{2}+\frac{1205315142}{35088329}a-\frac{5874635268}{1087738199}$, $\frac{4605641238}{1087738199}a^{15}-\frac{25454748111}{1087738199}a^{14}+\frac{38738629680}{1087738199}a^{13}+\frac{68519471539}{1087738199}a^{12}-\frac{312904867707}{1087738199}a^{11}+\frac{154169453681}{1087738199}a^{10}+\frac{20751056597}{35088329}a^{9}-\frac{562126419864}{1087738199}a^{8}-\frac{562119575973}{1087738199}a^{7}+\frac{327229967622}{1087738199}a^{6}+\frac{428214073737}{1087738199}a^{5}+\frac{103829579072}{1087738199}a^{4}-\frac{352123608056}{1087738199}a^{3}-\frac{4050452990}{1087738199}a^{2}+\frac{2228381743}{35088329}a-\frac{10963936090}{1087738199}$, $\frac{2449622833}{1087738199}a^{15}-\frac{13295083882}{1087738199}a^{14}+\frac{19358132186}{1087738199}a^{13}+\frac{38076449144}{1087738199}a^{12}-\frac{162662843246}{1087738199}a^{11}+\frac{67834274881}{1087738199}a^{10}+\frac{11150912994}{35088329}a^{9}-\frac{269446087103}{1087738199}a^{8}-\frac{313568611129}{1087738199}a^{7}+\frac{149395976157}{1087738199}a^{6}+\frac{227140911693}{1087738199}a^{5}+\frac{67933021142}{1087738199}a^{4}-\frac{174960013110}{1087738199}a^{3}-\frac{9275766765}{1087738199}a^{2}+\frac{1205315142}{35088329}a-\frac{4786897069}{1087738199}$, $\frac{6902337261}{1087738199}a^{15}-\frac{38239673763}{1087738199}a^{14}+\frac{58335764077}{1087738199}a^{13}+\frac{102808442309}{1087738199}a^{12}-\frac{470725535643}{1087738199}a^{11}+\frac{232996378197}{1087738199}a^{10}+\frac{31278318434}{35088329}a^{9}-\frac{847773158120}{1087738199}a^{8}-\frac{853426115632}{1087738199}a^{7}+\frac{491633146593}{1087738199}a^{6}+\frac{653574760308}{1087738199}a^{5}+\frac{161930661858}{1087738199}a^{4}-\frac{532020958466}{1087738199}a^{3}-\frac{10407793201}{1087738199}a^{2}+\frac{3378246731}{35088329}a-\frac{14332725507}{1087738199}$, $\frac{38675649}{35088329}a^{15}-\frac{220162298}{35088329}a^{14}+\frac{359599288}{35088329}a^{13}+\frac{528774725}{35088329}a^{12}-\frac{2737579388}{35088329}a^{11}+\frac{1717606926}{35088329}a^{10}+\frac{5288324002}{35088329}a^{9}-\frac{5718955480}{35088329}a^{8}-\frac{4107617906}{35088329}a^{7}+\frac{3845459058}{35088329}a^{6}+\frac{3242604861}{35088329}a^{5}+\frac{31672381}{35088329}a^{4}-\frac{3259758314}{35088329}a^{3}+\frac{460718207}{35088329}a^{2}+\frac{803487939}{35088329}a-\frac{190997351}{35088329}$, $\frac{3677848381}{1087738199}a^{15}-\frac{20907198456}{1087738199}a^{14}+\frac{33646511583}{1087738199}a^{13}+\frac{52062128271}{1087738199}a^{12}-\frac{260461989935}{1087738199}a^{11}+\frac{153597929173}{1087738199}a^{10}+\frac{16710409986}{35088329}a^{9}-\frac{522824387214}{1087738199}a^{8}-\frac{435991386668}{1087738199}a^{7}+\frac{331693323658}{1087738199}a^{6}+\frac{354674072356}{1087738199}a^{5}+\frac{48866439799}{1087738199}a^{4}-\frac{313208452392}{1087738199}a^{3}+\frac{13845233513}{1087738199}a^{2}+\frac{2007451996}{35088329}a-\frac{11667183225}{1087738199}$, $\frac{5804586357}{1087738199}a^{15}-\frac{32279779349}{1087738199}a^{14}+\frac{49886207608}{1087738199}a^{13}+\frac{84911488014}{1087738199}a^{12}-\frac{397769828735}{1087738199}a^{11}+\frac{207415268387}{1087738199}a^{10}+\frac{26039380599}{35088329}a^{9}-\frac{739414039744}{1087738199}a^{8}-\frac{689455731059}{1087738199}a^{7}+\frac{446439198420}{1087738199}a^{6}+\frac{528734824428}{1087738199}a^{5}+\frac{104811422883}{1087738199}a^{4}-\frac{453176115790}{1087738199}a^{3}+\frac{10231811427}{1087738199}a^{2}+\frac{2996781353}{35088329}a-\frac{16884853971}{1087738199}$, $\frac{8256880701}{1087738199}a^{15}-\frac{46140675083}{1087738199}a^{14}+\frac{71769254266}{1087738199}a^{13}+\frac{120636450915}{1087738199}a^{12}-\frac{569711870039}{1087738199}a^{11}+\frac{300389630846}{1087738199}a^{10}+\frac{37420606901}{35088329}a^{9}-\frac{1065599896893}{1087738199}a^{8}-\frac{1013219675975}{1087738199}a^{7}+\frac{646859800522}{1087738199}a^{6}+\frac{797849527562}{1087738199}a^{5}+\frac{159284275848}{1087738199}a^{4}-\frac{666085405450}{1087738199}a^{3}-\frac{7288312477}{1087738199}a^{2}+\frac{4337645335}{35088329}a-\frac{18653284480}{1087738199}$, $\frac{4586373514}{1087738199}a^{15}-\frac{25188868194}{1087738199}a^{14}+\frac{37520955486}{1087738199}a^{13}+\frac{70331213799}{1087738199}a^{12}-\frac{309837573412}{1087738199}a^{11}+\frac{139618556512}{1087738199}a^{10}+\frac{21087162014}{35088329}a^{9}-\frac{534569257064}{1087738199}a^{8}-\frac{597211541616}{1087738199}a^{7}+\frac{303897108097}{1087738199}a^{6}+\frac{452075402113}{1087738199}a^{5}+\frac{128785809649}{1087738199}a^{4}-\frac{350262991224}{1087738199}a^{3}-\frac{25914626086}{1087738199}a^{2}+\frac{2225226587}{35088329}a-\frac{7452439351}{1087738199}$, $\frac{6631813745}{1087738199}a^{15}-\frac{36868013819}{1087738199}a^{14}+\frac{56835049812}{1087738199}a^{13}+\frac{97250218308}{1087738199}a^{12}-\frac{453259919186}{1087738199}a^{11}+\frac{232739703073}{1087738199}a^{10}+\frac{29776711622}{35088329}a^{9}-\frac{827263042806}{1087738199}a^{8}-\frac{802149615345}{1087738199}a^{7}+\frac{480295846179}{1087738199}a^{6}+\frac{622376252447}{1087738199}a^{5}+\frac{147170461604}{1087738199}a^{4}-\frac{513097559200}{1087738199}a^{3}-\frac{5742452211}{1087738199}a^{2}+\frac{3156377881}{35088329}a-\frac{14151469676}{1087738199}$ Copy content Toggle raw display (assuming GRH)
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:  \( 23191.2132427 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{3}\cdot 23191.2132427 \cdot 1}{2\cdot\sqrt{298892216010498046875}}\cr\approx \mathstrut & 0.170363177961 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 11*x^14 + 11*x^13 - 75*x^12 + 65*x^11 + 125*x^10 - 187*x^9 - 68*x^8 + 128*x^7 + 63*x^6 - 20*x^5 - 88*x^4 + 33*x^3 + 16*x^2 - 9*x + 1)
 
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^16 - 6*x^15 + 11*x^14 + 11*x^13 - 75*x^12 + 65*x^11 + 125*x^10 - 187*x^9 - 68*x^8 + 128*x^7 + 63*x^6 - 20*x^5 - 88*x^4 + 33*x^3 + 16*x^2 - 9*x + 1, 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^16 - 6*x^15 + 11*x^14 + 11*x^13 - 75*x^12 + 65*x^11 + 125*x^10 - 187*x^9 - 68*x^8 + 128*x^7 + 63*x^6 - 20*x^5 - 88*x^4 + 33*x^3 + 16*x^2 - 9*x + 1);
 
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^16 - 6*x^15 + 11*x^14 + 11*x^13 - 75*x^12 + 65*x^11 + 125*x^10 - 187*x^9 - 68*x^8 + 128*x^7 + 63*x^6 - 20*x^5 - 88*x^4 + 33*x^3 + 16*x^2 - 9*x + 1);
 
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

$C_4^4.C_2\wr C_4$ (as 16T1771):

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 solvable group of order 16384
The 190 conjugacy class representatives for $C_4^4.C_2\wr C_4$
Character table for $C_4^4.C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.8.1292203125.1

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 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.8.52401279790283203125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $16$ R R $16$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ $16$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
\(179\) Copy content Toggle raw display $\Q_{179}$$x + 177$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 177$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 177$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 177$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 177$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 177$$1$$1$$0$Trivial$[\ ]$
179.2.1.1$x^{2} + 358$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.0.1$x^{2} + 172 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} + 172 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
179.4.0.1$x^{4} + x^{2} + 109 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
\(1021\) Copy content Toggle raw display $\Q_{1021}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1021}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$2$$2$$2$