Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^29 - 4*x - 4)
gp: K = bnfinit(y^29 - 4*y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 - 4*x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 - 4*x - 4)
\( x^{29} - 4x - 4 \)
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: |
\(26146730757746268604771485057902487410941420896256\)
\(\medspace = 2^{28}\cdot 53\cdot 179\cdot 10\!\cdots\!23\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(50.59\) | 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: | $2^{28/29}53^{1/2}179^{1/2}10267119471732676863546062197607495323^{1/2}\approx 6.094504598928962e+20$ | ||
| Ramified primes: |
\(2\), \(53\), \(179\), \(10267\!\cdots\!95323\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{97404\!\cdots\!29301}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{16}$, $\frac{1}{2}a^{17}$, $\frac{1}{2}a^{18}$, $\frac{1}{2}a^{19}$, $\frac{1}{2}a^{20}$, $\frac{1}{2}a^{21}$, $\frac{1}{2}a^{22}$, $\frac{1}{2}a^{23}$, $\frac{1}{2}a^{24}$, $\frac{1}{2}a^{25}$, $\frac{1}{2}a^{26}$, $\frac{1}{2}a^{27}$, $\frac{1}{10}a^{28}+\frac{1}{5}a^{27}-\frac{1}{10}a^{26}-\frac{1}{5}a^{25}+\frac{1}{10}a^{24}+\frac{1}{5}a^{23}-\frac{1}{10}a^{22}-\frac{1}{5}a^{21}+\frac{1}{10}a^{20}+\frac{1}{5}a^{19}-\frac{1}{10}a^{18}-\frac{1}{5}a^{17}+\frac{1}{10}a^{16}+\frac{1}{5}a^{15}+\frac{2}{5}a^{14}-\frac{1}{5}a^{13}-\frac{2}{5}a^{12}+\frac{1}{5}a^{11}+\frac{2}{5}a^{10}-\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$
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: |
$\frac{1}{2}a^{28}-\frac{1}{2}a^{27}+\frac{1}{2}a^{26}-\frac{1}{2}a^{25}+\frac{1}{2}a^{24}-\frac{1}{2}a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{21}+\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-3$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{27}+\frac{1}{2}a^{26}-\frac{1}{2}a^{25}+\frac{1}{2}a^{24}-\frac{1}{2}a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{21}+\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-1$, $\frac{1}{2}a^{22}+\frac{1}{2}a^{15}+a^{8}+2a+1$, $\frac{1}{10}a^{28}-\frac{3}{10}a^{27}-\frac{1}{10}a^{26}-\frac{1}{5}a^{25}+\frac{1}{10}a^{24}+\frac{1}{5}a^{23}-\frac{1}{10}a^{22}+\frac{3}{10}a^{21}+\frac{1}{10}a^{20}+\frac{1}{5}a^{19}+\frac{2}{5}a^{18}+\frac{3}{10}a^{17}-\frac{2}{5}a^{16}-\frac{3}{10}a^{15}-\frac{3}{5}a^{14}-\frac{1}{5}a^{13}-\frac{2}{5}a^{12}+\frac{1}{5}a^{11}-\frac{3}{5}a^{10}-\frac{1}{5}a^{9}+\frac{3}{5}a^{8}+\frac{1}{5}a^{7}+\frac{7}{5}a^{6}+\frac{4}{5}a^{5}+\frac{3}{5}a^{4}-\frac{4}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a-\frac{9}{5}$, $\frac{1}{10}a^{28}-\frac{3}{10}a^{27}+\frac{2}{5}a^{26}-\frac{7}{10}a^{25}+\frac{3}{5}a^{24}-\frac{3}{10}a^{23}+\frac{2}{5}a^{22}-\frac{7}{10}a^{21}+\frac{3}{5}a^{20}-\frac{3}{10}a^{19}-\frac{1}{10}a^{18}+\frac{3}{10}a^{17}+\frac{1}{10}a^{16}+\frac{1}{5}a^{15}-\frac{3}{5}a^{14}+\frac{4}{5}a^{13}-\frac{2}{5}a^{12}+\frac{1}{5}a^{11}-\frac{3}{5}a^{10}+\frac{4}{5}a^{9}-\frac{2}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{4}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$, $\frac{3}{10}a^{28}-\frac{2}{5}a^{27}-\frac{3}{10}a^{26}-\frac{1}{10}a^{25}-\frac{1}{5}a^{24}+\frac{3}{5}a^{23}-\frac{3}{10}a^{22}+\frac{2}{5}a^{21}-\frac{1}{5}a^{20}+\frac{3}{5}a^{19}-\frac{3}{10}a^{18}-\frac{1}{10}a^{17}-\frac{7}{10}a^{16}+\frac{1}{10}a^{15}+\frac{1}{5}a^{14}+\frac{2}{5}a^{13}-\frac{1}{5}a^{12}-\frac{2}{5}a^{11}+\frac{1}{5}a^{10}-\frac{3}{5}a^{9}-\frac{1}{5}a^{8}-\frac{7}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{9}{5}a^{4}+\frac{3}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{7}{5}$, $\frac{2}{5}a^{28}-\frac{1}{5}a^{27}+\frac{1}{10}a^{26}+\frac{1}{5}a^{25}+\frac{2}{5}a^{24}+\frac{3}{10}a^{23}-\frac{2}{5}a^{22}-\frac{3}{10}a^{21}-\frac{3}{5}a^{20}+\frac{3}{10}a^{19}+\frac{1}{10}a^{18}+\frac{1}{5}a^{17}-\frac{3}{5}a^{16}-\frac{7}{10}a^{15}-\frac{2}{5}a^{14}+\frac{1}{5}a^{13}+\frac{2}{5}a^{12}-\frac{1}{5}a^{11}-\frac{2}{5}a^{10}-\frac{4}{5}a^{9}+\frac{2}{5}a^{8}+\frac{4}{5}a^{7}+\frac{3}{5}a^{6}-\frac{4}{5}a^{5}-\frac{8}{5}a^{4}-\frac{6}{5}a^{3}+\frac{3}{5}a^{2}+\frac{11}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{28}-\frac{3}{5}a^{27}-\frac{1}{5}a^{26}+\frac{3}{5}a^{25}+\frac{1}{5}a^{24}-\frac{3}{5}a^{23}-\frac{1}{5}a^{22}+\frac{3}{5}a^{21}+\frac{1}{5}a^{20}-\frac{3}{5}a^{19}-\frac{1}{5}a^{18}+\frac{3}{5}a^{17}+\frac{1}{5}a^{16}-\frac{3}{5}a^{15}-\frac{1}{5}a^{14}+\frac{3}{5}a^{13}+\frac{1}{5}a^{12}-\frac{3}{5}a^{11}-\frac{6}{5}a^{10}+\frac{3}{5}a^{9}+\frac{6}{5}a^{8}-\frac{3}{5}a^{7}-\frac{6}{5}a^{6}+\frac{3}{5}a^{5}+\frac{11}{5}a^{4}-\frac{3}{5}a^{3}-\frac{11}{5}a^{2}+\frac{3}{5}a+\frac{7}{5}$, $\frac{9}{10}a^{28}-\frac{6}{5}a^{27}+\frac{3}{5}a^{26}-\frac{4}{5}a^{25}+\frac{7}{5}a^{24}-\frac{7}{10}a^{23}+\frac{11}{10}a^{22}-\frac{9}{5}a^{21}+\frac{9}{10}a^{20}-\frac{7}{10}a^{19}+\frac{8}{5}a^{18}-\frac{3}{10}a^{17}+\frac{2}{5}a^{16}-\frac{17}{10}a^{15}+\frac{3}{5}a^{14}+\frac{1}{5}a^{13}+\frac{7}{5}a^{12}-\frac{1}{5}a^{11}-\frac{2}{5}a^{10}-\frac{9}{5}a^{9}+\frac{7}{5}a^{8}+\frac{4}{5}a^{7}+\frac{8}{5}a^{6}-\frac{4}{5}a^{5}-\frac{8}{5}a^{4}-\frac{6}{5}a^{3}+\frac{8}{5}a^{2}+\frac{11}{5}a-\frac{11}{5}$, $\frac{1}{10}a^{28}+\frac{1}{5}a^{27}-\frac{1}{10}a^{26}-\frac{1}{5}a^{25}+\frac{3}{5}a^{24}-\frac{4}{5}a^{23}+\frac{2}{5}a^{22}-\frac{1}{5}a^{21}-\frac{2}{5}a^{20}-\frac{3}{10}a^{19}+\frac{2}{5}a^{18}-\frac{7}{10}a^{17}+\frac{3}{5}a^{16}+\frac{1}{5}a^{15}+\frac{2}{5}a^{14}+\frac{4}{5}a^{13}+\frac{3}{5}a^{12}+\frac{1}{5}a^{11}+\frac{7}{5}a^{10}-\frac{1}{5}a^{9}+\frac{3}{5}a^{8}+\frac{1}{5}a^{7}-\frac{3}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{9}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a-\frac{9}{5}$, $\frac{1}{2}a^{27}+\frac{1}{2}a^{25}-\frac{1}{2}a^{24}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-a^{16}-\frac{1}{2}a^{15}-a^{14}+a^{7}+a^{6}+2a^{5}+2a^{4}+a^{3}+a^{2}+1$, $\frac{3}{5}a^{28}-\frac{3}{10}a^{27}-\frac{1}{10}a^{26}-\frac{7}{10}a^{25}+\frac{3}{5}a^{24}-\frac{3}{10}a^{23}-\frac{3}{5}a^{22}+\frac{3}{10}a^{21}+\frac{3}{5}a^{20}-\frac{3}{10}a^{19}-\frac{1}{10}a^{18}+\frac{4}{5}a^{17}+\frac{3}{5}a^{16}-\frac{4}{5}a^{15}+\frac{2}{5}a^{14}+\frac{4}{5}a^{13}-\frac{2}{5}a^{12}-\frac{4}{5}a^{11}+\frac{2}{5}a^{10}+\frac{4}{5}a^{9}-\frac{7}{5}a^{8}+\frac{1}{5}a^{7}+\frac{7}{5}a^{6}-\frac{1}{5}a^{5}-\frac{7}{5}a^{4}+\frac{6}{5}a^{3}+\frac{7}{5}a^{2}-\frac{11}{5}a-\frac{19}{5}$, $\frac{1}{5}a^{28}-\frac{1}{10}a^{27}+\frac{3}{10}a^{26}-\frac{2}{5}a^{25}+\frac{1}{5}a^{24}-\frac{3}{5}a^{23}+\frac{3}{10}a^{22}+\frac{1}{10}a^{21}+\frac{1}{5}a^{20}-\frac{1}{10}a^{19}-\frac{1}{5}a^{18}+\frac{1}{10}a^{17}+\frac{1}{5}a^{16}-\frac{3}{5}a^{15}+\frac{4}{5}a^{14}-\frac{2}{5}a^{13}+\frac{1}{5}a^{12}-\frac{3}{5}a^{11}+\frac{4}{5}a^{10}+\frac{3}{5}a^{9}-\frac{4}{5}a^{8}-\frac{3}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{4}{5}a^{2}-\frac{2}{5}a-\frac{3}{5}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{26}+\frac{1}{2}a^{25}+\frac{1}{2}a^{24}-a^{23}+\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}+\frac{1}{2}a^{17}-\frac{1}{2}a^{16}+a^{14}-a^{12}+a^{10}-a^{9}-a^{8}+2a^{7}-2a^{5}+a^{4}+a^{3}-a^{2}-a-1$
| 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: | \( 28349352931361.156 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 28349352931361.156 \cdot 1}{2\cdot\sqrt{26146730757746268604771485057902487410941420896256}}\cr\approx \mathstrut & 0.828614655435449 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^29 - 4*x - 4)
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 - 4*x - 4, 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 - 4*x - 4);
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 - 4*x - 4);
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 | R | $22{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | $23{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $29$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $29$ | $21{,}\,{\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.14.0.1}{14} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $27{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/37.8.0.1}{8} }$ | $18{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.6.0.1}{6} }$ | $23{,}\,{\href{/padicField/47.6.0.1}{6} }$ | R | $20{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $29$ | $29$ | $1$ | $28$ | |||
|
\(53\)
| $\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 53.2.1.2 | $x^{2} + 106$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 53.8.0.1 | $x^{8} + 8 x^{4} + 29 x^{3} + 18 x^{2} + x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 53.17.0.1 | $x^{17} + 12 x + 51$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
|
\(179\)
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 179.2.1.1 | $x^{2} + 358$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 179.4.0.1 | $x^{4} + x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 179.5.0.1 | $x^{5} + 2 x + 177$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 179.17.0.1 | $x^{17} + 4 x + 177$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
|
\(102\!\cdots\!323\)
| $\Q_{10\!\cdots\!23}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{10\!\cdots\!23}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{10\!\cdots\!23}$ | $x$ | $1$ | $1$ | $0$ | 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |