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Properties

Label 16.10.724...000.1
Degree $16$
Signature $[10, 3]$
Discriminant $-7.249\times 10^{20}$
Root discriminant \(20.13\)
Ramified primes $2,5,191,761$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_4^4.C_2\wr C_2^2$ (as 16T1769)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 4*x^14 + 36*x^13 - 32*x^12 - 24*x^11 - 12*x^10 + 26*x^9 + 87*x^8 - 24*x^7 - 48*x^6 - 42*x^5 + 15*x^4 + 34*x^3 - x^2 - 6*x - 1)
 
gp: K = bnfinit(y^16 - 4*y^15 - 4*y^14 + 36*y^13 - 32*y^12 - 24*y^11 - 12*y^10 + 26*y^9 + 87*y^8 - 24*y^7 - 48*y^6 - 42*y^5 + 15*y^4 + 34*y^3 - y^2 - 6*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 - 4*x^14 + 36*x^13 - 32*x^12 - 24*x^11 - 12*x^10 + 26*x^9 + 87*x^8 - 24*x^7 - 48*x^6 - 42*x^5 + 15*x^4 + 34*x^3 - x^2 - 6*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 - 4*x^14 + 36*x^13 - 32*x^12 - 24*x^11 - 12*x^10 + 26*x^9 + 87*x^8 - 24*x^7 - 48*x^6 - 42*x^5 + 15*x^4 + 34*x^3 - x^2 - 6*x - 1)
 

\( x^{16} - 4 x^{15} - 4 x^{14} + 36 x^{13} - 32 x^{12} - 24 x^{11} - 12 x^{10} + 26 x^{9} + 87 x^{8} + \cdots - 1 \) 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:   \(-724907530649600000000\) \(\medspace = -\,2^{24}\cdot 5^{8}\cdot 191\cdot 761^{2}\) 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:  \(20.13\)
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:  $2^{3/2}5^{1/2}191^{1/2}761^{1/2}\approx 2411.232050218311$
Ramified primes:   \(2\), \(5\), \(191\), \(761\) 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:  \(\Q(\sqrt{-191}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{773672751}a^{15}+\frac{99984256}{773672751}a^{14}+\frac{319696264}{773672751}a^{13}-\frac{153824218}{773672751}a^{12}+\frac{160579745}{773672751}a^{11}-\frac{277435070}{773672751}a^{10}-\frac{371611621}{773672751}a^{9}-\frac{46834033}{257890917}a^{8}+\frac{18696377}{85963639}a^{7}-\frac{119711030}{257890917}a^{6}+\frac{49524832}{257890917}a^{5}+\frac{4173350}{85963639}a^{4}+\frac{7585748}{257890917}a^{3}+\frac{55997731}{773672751}a^{2}-\frac{58540211}{773672751}a+\frac{53989727}{773672751}$ 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:   $\frac{3441109264}{773672751}a^{15}-\frac{14737944722}{773672751}a^{14}-\frac{9598988972}{773672751}a^{13}+\frac{126602714777}{773672751}a^{12}-\frac{145867290874}{773672751}a^{11}-\frac{41489323697}{773672751}a^{10}-\frac{29769696376}{773672751}a^{9}+\frac{32921581448}{257890917}a^{8}+\frac{30114085882}{85963639}a^{7}-\frac{53084462699}{257890917}a^{6}-\frac{40312397378}{257890917}a^{5}-\frac{12265149195}{85963639}a^{4}+\frac{28082585936}{257890917}a^{3}+\frac{92849537491}{773672751}a^{2}-\frac{30241227044}{773672751}a-\frac{12451285585}{773672751}$, $a$, $\frac{2973629117}{773672751}a^{15}-\frac{12361996615}{773672751}a^{14}-\frac{9518568361}{773672751}a^{13}+\frac{107131068823}{773672751}a^{12}-\frac{114627777698}{773672751}a^{11}-\frac{40127585632}{773672751}a^{10}-\frac{34321811339}{773672751}a^{9}+\frac{24254080693}{257890917}a^{8}+\frac{25855914546}{85963639}a^{7}-\frac{36563546944}{257890917}a^{6}-\frac{31057150117}{257890917}a^{5}-\frac{10791853459}{85963639}a^{4}+\frac{19288032793}{257890917}a^{3}+\frac{74719730549}{773672751}a^{2}-\frac{21103436059}{773672751}a-\frac{9477656468}{773672751}$, $\frac{585711733}{85963639}a^{15}-\frac{2338032406}{85963639}a^{14}-\frac{2235470945}{85963639}a^{13}+\frac{20625621659}{85963639}a^{12}-\frac{19279453440}{85963639}a^{11}-\frac{10108143052}{85963639}a^{10}-\frac{9211148849}{85963639}a^{9}+\frac{11860822650}{85963639}a^{8}+\frac{47479667141}{85963639}a^{7}-\frac{12810285214}{85963639}a^{6}-\frac{17697782727}{85963639}a^{5}-\frac{22237865014}{85963639}a^{4}+\frac{5953123462}{85963639}a^{3}+\frac{13939728530}{85963639}a^{2}-\frac{2032466811}{85963639}a-\frac{1423495283}{85963639}$, $\frac{3558889207}{773672751}a^{15}-\frac{15791540402}{773672751}a^{14}-\frac{8240936426}{773672751}a^{13}+\frac{134859013868}{773672751}a^{12}-\frac{167637324448}{773672751}a^{11}-\frac{41405923925}{773672751}a^{10}-\frac{10832072221}{773672751}a^{9}+\frac{40464678860}{257890917}a^{8}+\frac{31547991655}{85963639}a^{7}-\frac{71504106074}{257890917}a^{6}-\frac{51203133554}{257890917}a^{5}-\frac{11559468390}{85963639}a^{4}+\frac{38725219832}{257890917}a^{3}+\frac{113967922546}{773672751}a^{2}-\frac{39771033920}{773672751}a-\frac{18532202899}{773672751}$, $\frac{835382990}{257890917}a^{15}-\frac{3586573753}{257890917}a^{14}-\frac{2372376499}{257890917}a^{13}+\frac{31025044042}{257890917}a^{12}-\frac{35274924767}{257890917}a^{11}-\frac{12148276438}{257890917}a^{10}-\frac{5994920675}{257890917}a^{9}+\frac{8548108567}{85963639}a^{8}+\frac{22715270517}{85963639}a^{7}-\frac{13042186349}{85963639}a^{6}-\frac{11565482902}{85963639}a^{5}-\frac{9490485001}{85963639}a^{4}+\frac{6936372110}{85963639}a^{3}+\frac{25310580794}{257890917}a^{2}-\frac{6309566611}{257890917}a-\frac{3715354619}{257890917}$, $\frac{2780244373}{257890917}a^{15}-\frac{10521928418}{257890917}a^{14}-\frac{12670858118}{257890917}a^{13}+\frac{94830571844}{257890917}a^{12}-\frac{72458694262}{257890917}a^{11}-\frac{58894387601}{257890917}a^{10}-\frac{58692998938}{257890917}a^{9}+\frac{13739819081}{85963639}a^{8}+\frac{77005797337}{85963639}a^{7}-\frac{3941546359}{85963639}a^{6}-\frac{25005027800}{85963639}a^{5}-\frac{40080769449}{85963639}a^{4}-\frac{36624996}{85963639}a^{3}+\frac{59667861511}{257890917}a^{2}+\frac{985424848}{257890917}a-\frac{3486015667}{257890917}$, $\frac{3082369418}{773672751}a^{15}-\frac{6803895166}{773672751}a^{14}-\frac{31088202481}{773672751}a^{13}+\frac{78266735530}{773672751}a^{12}+\frac{77885861689}{773672751}a^{11}-\frac{148253112379}{773672751}a^{10}-\frac{190239789662}{773672751}a^{9}-\frac{30218326001}{257890917}a^{8}+\frac{32103572102}{85963639}a^{7}+\frac{131997260888}{257890917}a^{6}+\frac{3200563856}{257890917}a^{5}-\frac{26264484141}{85963639}a^{4}-\frac{77892801284}{257890917}a^{3}+\frac{2937530213}{773672751}a^{2}+\frac{85090103189}{773672751}a+\frac{19697728099}{773672751}$, $\frac{2206513187}{773672751}a^{15}-\frac{9740182795}{773672751}a^{14}-\frac{4971692764}{773672751}a^{13}+\frac{82218818212}{773672751}a^{12}-\frac{103946349941}{773672751}a^{11}-\frac{16370436079}{773672751}a^{10}-\frac{14050846772}{773672751}a^{9}+\frac{22397718346}{257890917}a^{8}+\frac{18641055881}{85963639}a^{7}-\frac{42274546789}{257890917}a^{6}-\frac{23788993255}{257890917}a^{5}-\frac{6788208556}{85963639}a^{4}+\frac{22127021866}{257890917}a^{3}+\frac{58278024602}{773672751}a^{2}-\frac{25036711921}{773672751}a-\frac{8658480659}{773672751}$, $\frac{4879990072}{773672751}a^{15}-\frac{19837575146}{773672751}a^{14}-\frac{17395607528}{773672751}a^{13}+\frac{173949680315}{773672751}a^{12}-\frac{172315556785}{773672751}a^{11}-\frac{79009957547}{773672751}a^{10}-\frac{65687811868}{773672751}a^{9}+\frac{35984649395}{257890917}a^{8}+\frac{43743955836}{85963639}a^{7}-\frac{46460716703}{257890917}a^{6}-\frac{51104228939}{257890917}a^{5}-\frac{19475935579}{85963639}a^{4}+\frac{22328529047}{257890917}a^{3}+\frac{121448674540}{773672751}a^{2}-\frac{26493619373}{773672751}a-\frac{12328436308}{773672751}$, $\frac{10718809366}{773672751}a^{15}-\frac{45581221349}{773672751}a^{14}-\frac{31044759125}{773672751}a^{13}+\frac{392622356942}{773672751}a^{12}-\frac{443976607228}{773672751}a^{11}-\frac{135086879876}{773672751}a^{10}-\frac{99225414310}{773672751}a^{9}+\frac{98895212003}{257890917}a^{8}+\frac{94152132915}{85963639}a^{7}-\frac{157015365287}{257890917}a^{6}-\frac{123818421971}{257890917}a^{5}-\frac{38788521441}{85963639}a^{4}+\frac{82111122935}{257890917}a^{3}+\frac{289469083234}{773672751}a^{2}-\frac{87145795994}{773672751}a-\frac{37962459115}{773672751}$, $\frac{6219414218}{773672751}a^{15}-\frac{24318913432}{773672751}a^{14}-\frac{25420044472}{773672751}a^{13}+\frac{215868495604}{773672751}a^{12}-\frac{188684610383}{773672751}a^{11}-\frac{113009897950}{773672751}a^{10}-\frac{112790992691}{773672751}a^{9}+\frac{36946679608}{257890917}a^{8}+\frac{55967812075}{85963639}a^{7}-\frac{31385291515}{257890917}a^{6}-\frac{56540377294}{257890917}a^{5}-\frac{27226568424}{85963639}a^{4}+\frac{13219556719}{257890917}a^{3}+\frac{136229779982}{773672751}a^{2}-\frac{15745297183}{773672751}a-\frac{10869602102}{773672751}$ 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:  \( 43517.2044704 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{3}\cdot 43517.2044704 \cdot 1}{2\cdot\sqrt{724907530649600000000}}\cr\approx \mathstrut & 0.205271780341 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 4*x^14 + 36*x^13 - 32*x^12 - 24*x^11 - 12*x^10 + 26*x^9 + 87*x^8 - 24*x^7 - 48*x^6 - 42*x^5 + 15*x^4 + 34*x^3 - x^2 - 6*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 - 4*x^15 - 4*x^14 + 36*x^13 - 32*x^12 - 24*x^11 - 12*x^10 + 26*x^9 + 87*x^8 - 24*x^7 - 48*x^6 - 42*x^5 + 15*x^4 + 34*x^3 - x^2 - 6*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 - 4*x^15 - 4*x^14 + 36*x^13 - 32*x^12 - 24*x^11 - 12*x^10 + 26*x^9 + 87*x^8 - 24*x^7 - 48*x^6 - 42*x^5 + 15*x^4 + 34*x^3 - x^2 - 6*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 - 4*x^15 - 4*x^14 + 36*x^13 - 32*x^12 - 24*x^11 - 12*x^10 + 26*x^9 + 87*x^8 - 24*x^7 - 48*x^6 - 42*x^5 + 15*x^4 + 34*x^3 - x^2 - 6*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

$C_4^4.C_2\wr C_2^2$ (as 16T1769):

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 220 conjugacy class representatives for $C_4^4.C_2\wr C_2^2$
Character table for $C_4^4.C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.1948160000.1

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: 16.8.181941311897600000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ R ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 2.8.12.1$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(191\) Copy content Toggle raw display 191.2.0.1$x^{2} + 190 x + 19$$1$$2$$0$$C_2$$[\ ]^{2}$
191.2.0.1$x^{2} + 190 x + 19$$1$$2$$0$$C_2$$[\ ]^{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.0.1$x^{2} + 190 x + 19$$1$$2$$0$$C_2$$[\ ]^{2}$
191.2.0.1$x^{2} + 190 x + 19$$1$$2$$0$$C_2$$[\ ]^{2}$
191.2.0.1$x^{2} + 190 x + 19$$1$$2$$0$$C_2$$[\ ]^{2}$
191.4.0.1$x^{4} + 7 x^{2} + 100 x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
\(761\) Copy content Toggle raw display $\Q_{761}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{761}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{761}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{761}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$