Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^29 - 5*x - 5)
gp: K = bnfinit(y^29 - 5*y - 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 - 5*x - 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 - 5*x - 5)
\( x^{29} - 5x - 5 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $29$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(9942214263789125036236993858649470315293259918689727783203125\)
\(\medspace = 5^{28}\cdot 17\cdot 401\cdot 578483771\cdot 67676598636720696780695919103\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(126.87\) | 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: | $5^{28/29}17^{1/2}401^{1/2}578483771^{1/2}67676598636720696780695919103^{1/2}\approx 2.443594218128532e+21$ | ||
| Ramified primes: |
\(5\), \(17\), \(401\), \(578483771\), \(67676598636720696780695919103\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{26688\!\cdots\!24421}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{3}a^{28}+\frac{1}{3}a^{27}+\frac{1}{3}a^{26}+\frac{1}{3}a^{25}+\frac{1}{3}a^{24}+\frac{1}{3}a^{23}+\frac{1}{3}a^{22}+\frac{1}{3}a^{21}+\frac{1}{3}a^{20}+\frac{1}{3}a^{19}+\frac{1}{3}a^{18}+\frac{1}{3}a^{17}+\frac{1}{3}a^{16}+\frac{1}{3}a^{15}+\frac{1}{3}a^{14}+\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$
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: | $14$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$a+1$, $\frac{23}{3}a^{28}-\frac{22}{3}a^{27}+\frac{11}{3}a^{26}-\frac{1}{3}a^{25}-\frac{13}{3}a^{24}+\frac{14}{3}a^{23}-\frac{16}{3}a^{22}+\frac{14}{3}a^{21}-\frac{10}{3}a^{20}+\frac{14}{3}a^{19}-\frac{1}{3}a^{18}-\frac{4}{3}a^{17}+\frac{26}{3}a^{16}-\frac{34}{3}a^{15}+\frac{35}{3}a^{14}-\frac{19}{3}a^{13}-\frac{4}{3}a^{12}+\frac{26}{3}a^{11}-\frac{31}{3}a^{10}+\frac{29}{3}a^{9}-\frac{16}{3}a^{8}+\frac{8}{3}a^{7}-\frac{10}{3}a^{6}-\frac{1}{3}a^{5}+\frac{2}{3}a^{4}-\frac{34}{3}a^{3}+\frac{47}{3}a^{2}-\frac{55}{3}a-\frac{83}{3}$, $\frac{10}{3}a^{28}+\frac{1}{3}a^{27}+\frac{1}{3}a^{26}-\frac{5}{3}a^{25}+\frac{1}{3}a^{24}+\frac{1}{3}a^{23}-\frac{5}{3}a^{22}+\frac{7}{3}a^{21}-\frac{2}{3}a^{20}-\frac{2}{3}a^{19}-\frac{14}{3}a^{18}+\frac{13}{3}a^{17}-\frac{2}{3}a^{16}+\frac{7}{3}a^{15}-\frac{8}{3}a^{14}+\frac{13}{3}a^{13}-\frac{17}{3}a^{12}+\frac{4}{3}a^{11}-\frac{5}{3}a^{10}+\frac{7}{3}a^{9}-\frac{29}{3}a^{8}+\frac{10}{3}a^{7}+\frac{13}{3}a^{6}+\frac{22}{3}a^{5}-\frac{14}{3}a^{4}+\frac{7}{3}a^{3}-\frac{29}{3}a^{2}-\frac{29}{3}a-\frac{58}{3}$, $\frac{25}{3}a^{28}+\frac{55}{3}a^{27}-\frac{47}{3}a^{26}-\frac{38}{3}a^{25}-\frac{11}{3}a^{24}+\frac{76}{3}a^{23}+\frac{10}{3}a^{22}-\frac{35}{3}a^{21}-\frac{68}{3}a^{20}+\frac{37}{3}a^{19}+\frac{64}{3}a^{18}+\frac{34}{3}a^{17}-\frac{86}{3}a^{16}-\frac{62}{3}a^{15}+\frac{40}{3}a^{14}+\frac{130}{3}a^{13}+\frac{4}{3}a^{12}-\frac{134}{3}a^{11}-\frac{95}{3}a^{10}+\frac{133}{3}a^{9}+\frac{169}{3}a^{8}-\frac{65}{3}a^{7}-\frac{248}{3}a^{6}-\frac{35}{3}a^{5}+\frac{268}{3}a^{4}+\frac{166}{3}a^{3}-\frac{248}{3}a^{2}-\frac{314}{3}a+\frac{2}{3}$, $18a^{28}-51a^{27}+21a^{26}-6a^{25}-4a^{24}+70a^{23}-23a^{22}+30a^{21}-88a^{19}+13a^{18}-60a^{17}+14a^{16}+83a^{15}-8a^{14}+109a^{13}-21a^{12}-76a^{11}+16a^{10}-140a^{9}+12a^{8}+23a^{7}-19a^{6}+163a^{5}-28a^{4}+67a^{3}+79a^{2}-166a-66$, $\frac{127}{3}a^{28}+\frac{403}{3}a^{27}+\frac{280}{3}a^{26}-\frac{242}{3}a^{25}-\frac{512}{3}a^{24}-\frac{263}{3}a^{23}+\frac{376}{3}a^{22}+\frac{646}{3}a^{21}+\frac{205}{3}a^{20}-\frac{554}{3}a^{19}-\frac{779}{3}a^{18}-\frac{110}{3}a^{17}+\frac{796}{3}a^{16}+\frac{898}{3}a^{15}-\frac{47}{3}a^{14}-\frac{1091}{3}a^{13}-\frac{1010}{3}a^{12}+\frac{307}{3}a^{11}+\frac{1435}{3}a^{10}+\frac{1090}{3}a^{9}-\frac{674}{3}a^{8}-\frac{1853}{3}a^{7}-\frac{1076}{3}a^{6}+\frac{1153}{3}a^{5}+\frac{2359}{3}a^{4}+\frac{931}{3}a^{3}-\frac{1784}{3}a^{2}-\frac{2903}{3}a-\frac{1282}{3}$, $\frac{362}{3}a^{28}-\frac{247}{3}a^{27}+\frac{107}{3}a^{26}-\frac{7}{3}a^{25}-\frac{82}{3}a^{24}+\frac{197}{3}a^{23}-\frac{328}{3}a^{22}+\frac{431}{3}a^{21}-\frac{460}{3}a^{20}+\frac{467}{3}a^{19}-\frac{472}{3}a^{18}+\frac{407}{3}a^{17}-\frac{304}{3}a^{16}+\frac{218}{3}a^{15}-\frac{115}{3}a^{14}-\frac{22}{3}a^{13}+\frac{215}{3}a^{12}-\frac{382}{3}a^{11}+\frac{416}{3}a^{10}-\frac{430}{3}a^{9}+\frac{503}{3}a^{8}-\frac{562}{3}a^{7}+\frac{584}{3}a^{6}-\frac{457}{3}a^{5}+\frac{212}{3}a^{4}-\frac{73}{3}a^{3}+\frac{38}{3}a^{2}+\frac{11}{3}a-\frac{2012}{3}$, $\frac{173}{3}a^{28}-\frac{31}{3}a^{27}-\frac{262}{3}a^{26}-\frac{400}{3}a^{25}-\frac{352}{3}a^{24}-\frac{133}{3}a^{23}+\frac{221}{3}a^{22}+\frac{551}{3}a^{21}+\frac{704}{3}a^{20}+\frac{593}{3}a^{19}+\frac{203}{3}a^{18}-\frac{313}{3}a^{17}-\frac{757}{3}a^{16}-\frac{919}{3}a^{15}-\frac{655}{3}a^{14}-\frac{70}{3}a^{13}+\frac{644}{3}a^{12}+\frac{1169}{3}a^{11}+\frac{1172}{3}a^{10}+\frac{608}{3}a^{9}-\frac{406}{3}a^{8}-\frac{1489}{3}a^{7}-\frac{2125}{3}a^{6}-\frac{1993}{3}a^{5}-\frac{976}{3}a^{4}+\frac{575}{3}a^{3}+\frac{2054}{3}a^{2}+\frac{2810}{3}a+\frac{1462}{3}$, $85a^{28}+98a^{27}+37a^{26}+50a^{25}-38a^{24}-73a^{23}-124a^{22}-136a^{21}-173a^{20}-90a^{19}-65a^{18}+19a^{17}+154a^{16}+177a^{15}+272a^{14}+270a^{13}+210a^{12}+97a^{11}+3a^{10}-247a^{9}-322a^{8}-442a^{7}-509a^{6}-358a^{5}-231a^{4}+8a^{3}+343a^{2}+598a+301$, $60a^{28}-57a^{27}+45a^{26}-27a^{25}+2a^{24}+32a^{23}-67a^{22}+88a^{21}-77a^{20}+38a^{19}+7a^{18}-38a^{17}+58a^{16}-80a^{15}+96a^{14}-87a^{13}+38a^{12}+30a^{11}-93a^{10}+115a^{9}-105a^{8}+73a^{7}-54a^{6}+21a^{5}+37a^{4}-128a^{3}+176a^{2}-151a-236$, $\frac{37}{3}a^{28}+\frac{37}{3}a^{27}+\frac{25}{3}a^{26}-\frac{23}{3}a^{25}-\frac{35}{3}a^{24}-\frac{65}{3}a^{23}-\frac{20}{3}a^{22}-\frac{20}{3}a^{21}+\frac{19}{3}a^{20}-\frac{29}{3}a^{19}-\frac{44}{3}a^{18}-\frac{101}{3}a^{17}-\frac{71}{3}a^{16}-\frac{29}{3}a^{15}+\frac{55}{3}a^{14}+\frac{91}{3}a^{13}+\frac{73}{3}a^{12}-\frac{2}{3}a^{11}-\frac{80}{3}a^{10}-\frac{83}{3}a^{9}-\frac{14}{3}a^{8}+\frac{106}{3}a^{7}+\frac{187}{3}a^{6}+\frac{169}{3}a^{5}+\frac{70}{3}a^{4}-\frac{44}{3}a^{3}-\frac{35}{3}a^{2}+\frac{70}{3}a+\frac{98}{3}$, $\frac{77}{3}a^{28}-\frac{97}{3}a^{27}+\frac{71}{3}a^{26}-\frac{64}{3}a^{25}+\frac{110}{3}a^{24}-\frac{49}{3}a^{23}+\frac{80}{3}a^{22}-\frac{76}{3}a^{21}+\frac{47}{3}a^{20}-\frac{76}{3}a^{19}+\frac{77}{3}a^{18}-\frac{16}{3}a^{17}+\frac{101}{3}a^{16}-\frac{16}{3}a^{15}+\frac{38}{3}a^{14}-\frac{88}{3}a^{13}+\frac{8}{3}a^{12}-\frac{13}{3}a^{11}+\frac{107}{3}a^{10}+\frac{71}{3}a^{9}+\frac{74}{3}a^{8}-\frac{52}{3}a^{7}-\frac{43}{3}a^{6}-\frac{37}{3}a^{5}+\frac{32}{3}a^{4}+\frac{89}{3}a^{3}+\frac{161}{3}a^{2}+\frac{128}{3}a-\frac{377}{3}$, $\frac{193}{3}a^{28}+\frac{178}{3}a^{27}-\frac{305}{3}a^{26}-\frac{68}{3}a^{25}+\frac{283}{3}a^{24}+\frac{103}{3}a^{23}-\frac{173}{3}a^{22}-\frac{281}{3}a^{21}+\frac{376}{3}a^{20}+\frac{304}{3}a^{19}-\frac{428}{3}a^{18}-\frac{200}{3}a^{17}+\frac{376}{3}a^{16}+\frac{358}{3}a^{15}-\frac{287}{3}a^{14}-\frac{575}{3}a^{13}+\frac{592}{3}a^{12}+\frac{589}{3}a^{11}-\frac{572}{3}a^{10}-\frac{464}{3}a^{9}+\frac{442}{3}a^{8}+\frac{868}{3}a^{7}-\frac{419}{3}a^{6}-\frac{1100}{3}a^{5}+\frac{892}{3}a^{4}+\frac{1036}{3}a^{3}-\frac{602}{3}a^{2}-\frac{1049}{3}a-\frac{502}{3}$, $\frac{1877}{3}a^{28}+\frac{287}{3}a^{27}-\frac{2029}{3}a^{26}+\frac{617}{3}a^{25}+\frac{2285}{3}a^{24}-\frac{1495}{3}a^{23}-\frac{1786}{3}a^{22}+\frac{2573}{3}a^{21}+\frac{1304}{3}a^{20}-\frac{3262}{3}a^{19}+\frac{251}{3}a^{18}+\frac{3899}{3}a^{17}-\frac{1522}{3}a^{16}-\frac{3454}{3}a^{15}+\frac{3614}{3}a^{14}+\frac{2714}{3}a^{13}-\frac{4756}{3}a^{12}-\frac{721}{3}a^{11}+\frac{6311}{3}a^{10}-\frac{1099}{3}a^{9}-\frac{6022}{3}a^{8}+\frac{4388}{3}a^{7}+\frac{5957}{3}a^{6}-\frac{6694}{3}a^{5}-\frac{3181}{3}a^{4}+\frac{9767}{3}a^{3}+\frac{449}{3}a^{2}-\frac{10174}{3}a-\frac{4247}{3}$
| 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: | \( 13281089874686380000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 13281089874686380000 \cdot 1}{2\cdot\sqrt{9942214263789125036236993858649470315293259918689727783203125}}\cr\approx \mathstrut & 0.629521132323005 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^29 - 5*x - 5)
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^29 - 5*x - 5, 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^29 - 5*x - 5);
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^29 - 5*x - 5);
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
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 8841761993739701954543616000000 |
| The 4565 conjugacy class representatives for $S_{29}$ are not computed |
| Character table for $S_{29}$ is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $27{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | $18{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | $21{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $26{,}\,{\href{/padicField/13.3.0.1}{3} }$ | R | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $17{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $27{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $25{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $29$ | $16{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | $26{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $29$ | $29$ | $1$ | $28$ | |||
|
\(17\)
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.23.0.1 | $x^{23} + 15 x^{2} + 16 x + 14$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | |
|
\(401\)
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
|
\(578483771\)
| $\Q_{578483771}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(676\!\cdots\!103\)
| $\Q_{67\!\cdots\!03}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |