Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 2*x + 5)
gp: K = bnfinit(y^30 - 2*y + 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 2*x + 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 2*x + 5)
\( x^{30} - 2x + 5 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $30$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-9587552961643515075421179460717112868282985013768744056322523136\)
\(\medspace = -\,2^{31}\cdot 27697516504717\cdot 16\!\cdots\!21\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(135.74\) | 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: |
\(2\), \(27697516504717\), \(16118\!\cdots\!92721\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-89291\!\cdots\!29914}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{2}a^{29}-\frac{1}{2}a^{28}-\frac{1}{2}a^{27}-\frac{1}{2}a^{26}-\frac{1}{2}a^{25}-\frac{1}{2}a^{24}-\frac{1}{2}a^{23}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$
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: |
$\frac{5}{2}a^{29}-\frac{3}{2}a^{28}+\frac{1}{2}a^{27}+\frac{1}{2}a^{26}-\frac{3}{2}a^{25}+\frac{5}{2}a^{24}-\frac{7}{2}a^{23}+\frac{9}{2}a^{22}-\frac{11}{2}a^{21}+\frac{13}{2}a^{20}-\frac{15}{2}a^{19}+\frac{17}{2}a^{18}-\frac{19}{2}a^{17}+\frac{21}{2}a^{16}-\frac{23}{2}a^{15}+\frac{25}{2}a^{14}-\frac{27}{2}a^{13}+\frac{29}{2}a^{12}-\frac{31}{2}a^{11}+\frac{33}{2}a^{10}-\frac{35}{2}a^{9}+\frac{37}{2}a^{8}-\frac{39}{2}a^{7}+\frac{41}{2}a^{6}-\frac{43}{2}a^{5}+\frac{45}{2}a^{4}-\frac{47}{2}a^{3}+\frac{49}{2}a^{2}-\frac{49}{2}a+\frac{33}{2}$, $\frac{43}{2}a^{29}-\frac{27}{2}a^{28}-\frac{105}{2}a^{27}-\frac{145}{2}a^{26}-\frac{133}{2}a^{25}-\frac{65}{2}a^{24}+\frac{27}{2}a^{23}+\frac{91}{2}a^{22}+\frac{119}{2}a^{21}+\frac{115}{2}a^{20}+\frac{77}{2}a^{19}+\frac{47}{2}a^{18}+\frac{47}{2}a^{17}+\frac{13}{2}a^{16}-\frac{81}{2}a^{15}-\frac{171}{2}a^{14}-\frac{217}{2}a^{13}-\frac{201}{2}a^{12}-\frac{85}{2}a^{11}+\frac{97}{2}a^{10}+\frac{239}{2}a^{9}+\frac{285}{2}a^{8}+\frac{221}{2}a^{7}+\frac{73}{2}a^{6}-\frac{33}{2}a^{5}-\frac{59}{2}a^{4}-\frac{111}{2}a^{3}-\frac{179}{2}a^{2}-\frac{203}{2}a-\frac{339}{2}$, $\frac{91}{2}a^{29}+\frac{5}{2}a^{28}-\frac{85}{2}a^{27}-\frac{5}{2}a^{26}-\frac{3}{2}a^{25}-\frac{137}{2}a^{24}-\frac{69}{2}a^{23}+\frac{83}{2}a^{22}-\frac{1}{2}a^{21}-\frac{37}{2}a^{20}+\frac{147}{2}a^{19}+\frac{143}{2}a^{18}-\frac{37}{2}a^{17}+\frac{5}{2}a^{16}+\frac{89}{2}a^{15}-\frac{85}{2}a^{14}-\frac{191}{2}a^{13}-\frac{45}{2}a^{12}-\frac{1}{2}a^{11}-\frac{113}{2}a^{10}-\frac{47}{2}a^{9}+\frac{171}{2}a^{8}+\frac{177}{2}a^{7}-\frac{3}{2}a^{6}+\frac{77}{2}a^{5}+\frac{251}{2}a^{4}-\frac{49}{2}a^{3}-\frac{317}{2}a^{2}+\frac{7}{2}a-\frac{141}{2}$, $19a^{29}+88a^{28}-17a^{27}-97a^{26}+19a^{25}+105a^{24}-19a^{23}-122a^{22}+19a^{21}+142a^{20}-17a^{19}-154a^{18}+7a^{17}+173a^{16}-8a^{15}-190a^{14}+10a^{13}+214a^{12}-250a^{10}-7a^{9}+274a^{8}+21a^{7}-297a^{6}-32a^{5}+334a^{4}+34a^{3}-380a^{2}-46a+386$, $\frac{69}{2}a^{29}-\frac{143}{2}a^{28}+\frac{105}{2}a^{27}-\frac{27}{2}a^{26}+\frac{111}{2}a^{25}-\frac{91}{2}a^{24}-\frac{47}{2}a^{23}+\frac{27}{2}a^{22}-\frac{87}{2}a^{21}+\frac{301}{2}a^{20}-\frac{287}{2}a^{19}+\frac{229}{2}a^{18}-\frac{331}{2}a^{17}+\frac{191}{2}a^{16}+\frac{33}{2}a^{15}+\frac{31}{2}a^{14}+\frac{53}{2}a^{13}-\frac{271}{2}a^{12}+\frac{163}{2}a^{11}-\frac{193}{2}a^{10}+\frac{507}{2}a^{9}-\frac{469}{2}a^{8}+\frac{413}{2}a^{7}-\frac{637}{2}a^{6}+\frac{379}{2}a^{5}+\frac{85}{2}a^{4}-\frac{119}{2}a^{3}+\frac{451}{2}a^{2}-\frac{937}{2}a+\frac{579}{2}$, $\frac{91}{2}a^{29}-\frac{165}{2}a^{28}+\frac{71}{2}a^{27}+\frac{205}{2}a^{26}-\frac{467}{2}a^{25}+\frac{431}{2}a^{24}-\frac{115}{2}a^{23}-\frac{195}{2}a^{22}+\frac{251}{2}a^{21}-\frac{127}{2}a^{20}+\frac{137}{2}a^{19}-\frac{371}{2}a^{18}+\frac{571}{2}a^{17}-\frac{329}{2}a^{16}-\frac{317}{2}a^{15}+\frac{847}{2}a^{14}-\frac{839}{2}a^{13}+\frac{331}{2}a^{12}+\frac{67}{2}a^{11}-\frac{49}{2}a^{10}-\frac{81}{2}a^{9}-\frac{213}{2}a^{8}+\frac{977}{2}a^{7}-\frac{1463}{2}a^{6}+\frac{983}{2}a^{5}+\frac{283}{2}a^{4}-\frac{1301}{2}a^{3}+\frac{1319}{2}a^{2}-\frac{707}{2}a+\frac{203}{2}$, $27a^{29}+58a^{28}+6a^{27}-28a^{26}+19a^{25}+14a^{24}-74a^{23}-47a^{22}+a^{21}-43a^{20}-112a^{19}+3a^{18}+10a^{17}-52a^{16}-50a^{15}+116a^{14}+25a^{13}-15a^{12}+66a^{11}+181a^{10}-28a^{9}+14a^{8}+123a^{7}+87a^{6}-159a^{5}+15a^{4}+71a^{3}-127a^{2}-238a+4$, $90a^{29}+100a^{28}+98a^{27}+85a^{26}+24a^{25}-65a^{24}-108a^{23}-120a^{22}-129a^{21}-106a^{20}-12a^{19}+80a^{18}+136a^{17}+182a^{16}+187a^{15}+112a^{14}+10a^{13}-48a^{12}-142a^{11}-240a^{10}-229a^{9}-110a^{8}-35a^{7}+27a^{6}+174a^{5}+280a^{4}+262a^{3}+190a^{2}+129a-241$, $47a^{29}+20a^{28}-46a^{27}-114a^{26}-124a^{25}-147a^{24}-82a^{23}-38a^{22}+67a^{21}+112a^{20}+185a^{19}+182a^{18}+89a^{17}+107a^{16}-121a^{15}-115a^{14}-208a^{13}-286a^{12}-63a^{11}-201a^{10}+142a^{9}+185a^{8}+172a^{7}+449a^{6}+75a^{5}+225a^{4}-77a^{3}-304a^{2}-167a-647$, $\frac{83}{2}a^{29}+\frac{109}{2}a^{28}+\frac{213}{2}a^{27}-\frac{145}{2}a^{26}-\frac{245}{2}a^{25}-\frac{273}{2}a^{24}+\frac{135}{2}a^{23}-\frac{53}{2}a^{22}+\frac{137}{2}a^{21}+\frac{103}{2}a^{20}+\frac{497}{2}a^{19}-\frac{113}{2}a^{18}-\frac{217}{2}a^{17}-\frac{479}{2}a^{16}+\frac{139}{2}a^{15}-\frac{229}{2}a^{14}+\frac{51}{2}a^{13}+\frac{77}{2}a^{12}+\frac{803}{2}a^{11}+\frac{113}{2}a^{10}-\frac{225}{2}a^{9}-\frac{539}{2}a^{8}+\frac{51}{2}a^{7}-\frac{315}{2}a^{6}-\frac{393}{2}a^{5}+\frac{117}{2}a^{4}+\frac{933}{2}a^{3}+\frac{681}{2}a^{2}-\frac{401}{2}a-\frac{383}{2}$, $103a^{29}-89a^{28}+114a^{27}+252a^{26}-36a^{25}-82a^{24}+246a^{23}+131a^{22}-221a^{21}+30a^{20}+294a^{19}-115a^{18}-257a^{17}+231a^{16}+154a^{15}-408a^{14}-150a^{13}+325a^{12}-207a^{11}-563a^{10}+165a^{9}+238a^{8}-650a^{7}-396a^{6}+436a^{5}-260a^{4}-941a^{3}+88a^{2}+418a-1057$, $33a^{29}+41a^{28}-71a^{27}+13a^{26}+71a^{25}-75a^{24}-14a^{23}+96a^{22}-66a^{21}-52a^{20}+120a^{19}-40a^{18}-104a^{17}+135a^{16}+2a^{15}-152a^{14}+125a^{13}+70a^{12}-202a^{11}+89a^{10}+155a^{9}-232a^{8}+24a^{7}+236a^{6}-221a^{5}-86a^{4}+322a^{3}-175a^{2}-212a+307$, $\frac{1}{2}a^{29}-\frac{17}{2}a^{28}+\frac{19}{2}a^{27}+\frac{83}{2}a^{26}+\frac{29}{2}a^{25}+\frac{43}{2}a^{24}+\frac{81}{2}a^{23}+\frac{1}{2}a^{22}+\frac{59}{2}a^{21}+\frac{125}{2}a^{20}+\frac{41}{2}a^{19}+\frac{141}{2}a^{18}+\frac{217}{2}a^{17}+\frac{93}{2}a^{16}+\frac{157}{2}a^{15}+\frac{167}{2}a^{14}+\frac{33}{2}a^{13}+\frac{159}{2}a^{12}+\frac{195}{2}a^{11}+\frac{93}{2}a^{10}+\frac{237}{2}a^{9}+\frac{231}{2}a^{8}+\frac{133}{2}a^{7}+\frac{203}{2}a^{6}+\frac{63}{2}a^{5}+\frac{15}{2}a^{4}+\frac{153}{2}a^{3}+\frac{35}{2}a^{2}+\frac{111}{2}a+\frac{191}{2}$, $234a^{29}-259a^{28}+309a^{27}-330a^{26}+373a^{25}-398a^{24}+431a^{23}-460a^{22}+485a^{21}-497a^{20}+525a^{19}-508a^{18}+545a^{17}-490a^{16}+517a^{15}-437a^{14}+440a^{13}-358a^{12}+314a^{11}-212a^{10}+117a^{9}+11a^{8}-113a^{7}+282a^{6}-398a^{5}+639a^{4}-784a^{3}+1041a^{2}-1210a+976$
| 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: | \( 316128619265654000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 316128619265654000000 \cdot 1}{2\cdot\sqrt{9587552961643515075421179460717112868282985013768744056322523136}}\cr\approx \mathstrut & 1.51592584114682 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 - 2*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^30 - 2*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^30 - 2*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^30 - 2*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 265252859812191058636308480000000 |
| The 5604 conjugacy class representatives for $S_{30}$ are not computed |
| Character table for $S_{30}$ |
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 | R | $30$ | ${\href{/padicField/5.14.0.1}{14} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $27{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $18{,}\,{\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.14.0.1}{14} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $27{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | $20{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.4.4.4 | $x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ | |
| 2.8.8.7 | $x^{8} + 8 x^{7} + 40 x^{6} + 120 x^{5} + 232 x^{4} + 240 x^{3} + 160 x^{2} + 64 x + 16$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ | |
| 2.8.8.8 | $x^{8} - 6 x^{7} + 40 x^{6} - 52 x^{5} + 192 x^{4} + 328 x^{3} + 1376 x^{2} + 1264 x + 1328$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ | |
| 2.8.8.6 | $x^{8} + 2 x^{7} + 24 x^{6} + 84 x^{5} + 264 x^{4} + 408 x^{3} + 384 x^{2} - 208 x + 80$ | $2$ | $4$ | $8$ | $(C_8:C_2):C_2$ | $[2, 2, 2]^{4}$ | |
|
\(27697516504717\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ | ||
|
\(161\!\cdots\!721\)
| $\Q_{16\!\cdots\!21}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{16\!\cdots\!21}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ |