Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 11*x^18 - 11*x^17 - 2*x^16 + 16*x^15 - 2*x^14 - 24*x^13 + 22*x^12 + 8*x^11 - 22*x^10 - 13*x^9 + 14*x^8 + 3*x^7 + x^6 + 15*x^5 + 20*x^4 + 9*x^3 + 3*x^2 + 2*x + 1)
gp: K = bnfinit(y^20 - 5*y^19 + 11*y^18 - 11*y^17 - 2*y^16 + 16*y^15 - 2*y^14 - 24*y^13 + 22*y^12 + 8*y^11 - 22*y^10 - 13*y^9 + 14*y^8 + 3*y^7 + y^6 + 15*y^5 + 20*y^4 + 9*y^3 + 3*y^2 + 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^19 + 11*x^18 - 11*x^17 - 2*x^16 + 16*x^15 - 2*x^14 - 24*x^13 + 22*x^12 + 8*x^11 - 22*x^10 - 13*x^9 + 14*x^8 + 3*x^7 + x^6 + 15*x^5 + 20*x^4 + 9*x^3 + 3*x^2 + 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^19 + 11*x^18 - 11*x^17 - 2*x^16 + 16*x^15 - 2*x^14 - 24*x^13 + 22*x^12 + 8*x^11 - 22*x^10 - 13*x^9 + 14*x^8 + 3*x^7 + x^6 + 15*x^5 + 20*x^4 + 9*x^3 + 3*x^2 + 2*x + 1)
\( x^{20} - 5 x^{19} + 11 x^{18} - 11 x^{17} - 2 x^{16} + 16 x^{15} - 2 x^{14} - 24 x^{13} + 22 x^{12} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2199140361717170013184\)
\(\medspace = 2^{10}\cdot 7^{6}\cdot 23^{2}\cdot 431^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.67\) | 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: | $2^{15/8}7^{3/4}23^{1/2}431^{1/2}\approx 1571.6553852184147$ | ||
| Ramified primes: |
\(2\), \(7\), \(23\), \(431\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{567947447479}a^{19}-\frac{188290224203}{567947447479}a^{18}-\frac{174225288284}{567947447479}a^{17}+\frac{12253238385}{567947447479}a^{16}+\frac{18966534861}{567947447479}a^{15}-\frac{138182773363}{567947447479}a^{14}-\frac{257287552850}{567947447479}a^{13}-\frac{2825698004}{567947447479}a^{12}+\frac{214722052102}{567947447479}a^{11}+\frac{50164800310}{567947447479}a^{10}-\frac{152349283644}{567947447479}a^{9}+\frac{176256545782}{567947447479}a^{8}-\frac{30458889122}{567947447479}a^{7}-\frac{67544896520}{567947447479}a^{6}-\frac{129936515290}{567947447479}a^{5}+\frac{41891468240}{567947447479}a^{4}+\frac{13141853506}{567947447479}a^{3}-\frac{245090998797}{567947447479}a^{2}-\frac{238123711600}{567947447479}a-\frac{249453842442}{567947447479}$
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 $1$
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: | $9$ | 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: |
$\frac{68936272252}{567947447479}a^{19}-\frac{587143183448}{567947447479}a^{18}+\frac{2117265143094}{567947447479}a^{17}-\frac{4183267805908}{567947447479}a^{16}+\frac{4254724686062}{567947447479}a^{15}-\frac{231585731999}{567947447479}a^{14}-\frac{4207437433960}{567947447479}a^{13}+\frac{1419984163961}{567947447479}a^{12}+\frac{6742805600484}{567947447479}a^{11}-\frac{8950229774136}{567947447479}a^{10}+\frac{1365047554024}{567947447479}a^{9}+\frac{4662797225093}{567947447479}a^{8}+\frac{19788497056}{567947447479}a^{7}-\frac{3563299676417}{567947447479}a^{6}+\frac{1726772640216}{567947447479}a^{5}-\frac{163034697879}{567947447479}a^{4}-\frac{1236450365418}{567947447479}a^{3}-\frac{1891679423009}{567947447479}a^{2}+\frac{309646581486}{567947447479}a+\frac{43368248979}{567947447479}$, $\frac{293504572591}{567947447479}a^{19}-\frac{1704812644334}{567947447479}a^{18}+\frac{4336290135033}{567947447479}a^{17}-\frac{5222424524662}{567947447479}a^{16}-\frac{150813449678}{567947447479}a^{15}+\frac{9617111611187}{567947447479}a^{14}-\frac{9701653029943}{567947447479}a^{13}-\frac{3889288738888}{567947447479}a^{12}+\frac{13336426188188}{567947447479}a^{11}-\frac{3530194849954}{567947447479}a^{10}-\frac{12507855940696}{567947447479}a^{9}+\frac{8178727076943}{567947447479}a^{8}+\frac{3620815707770}{567947447479}a^{7}-\frac{2374922844801}{567947447479}a^{6}-\frac{1590117886975}{567947447479}a^{5}+\frac{5919555989051}{567947447479}a^{4}+\frac{955832415281}{567947447479}a^{3}-\frac{1307518715679}{567947447479}a^{2}-\frac{394787649623}{567947447479}a+\frac{662156881518}{567947447479}$, $\frac{158365982752}{567947447479}a^{19}-\frac{1241622898876}{567947447479}a^{18}+\frac{4404719589920}{567947447479}a^{17}-\frac{9150120090384}{567947447479}a^{16}+\frac{11425280313166}{567947447479}a^{15}-\frac{6963727576851}{567947447479}a^{14}-\frac{312340286963}{567947447479}a^{13}+\frac{72714588246}{567947447479}a^{12}+\frac{7776098937862}{567947447479}a^{11}-\frac{12314697346966}{567947447479}a^{10}+\frac{7479283188305}{567947447479}a^{9}-\frac{2417325969156}{567947447479}a^{8}+\frac{4757696478615}{567947447479}a^{7}-\frac{4288327603196}{567947447479}a^{6}+\frac{3202302708368}{567947447479}a^{5}-\frac{1622823035296}{567947447479}a^{4}-\frac{1285804128322}{567947447479}a^{3}-\frac{2578126148794}{567947447479}a^{2}-\frac{508076004183}{567947447479}a-\frac{766903201165}{567947447479}$, $\frac{487660646241}{567947447479}a^{19}-\frac{2714458694554}{567947447479}a^{18}+\frac{6978391276066}{567947447479}a^{17}-\frac{9653562081508}{567947447479}a^{16}+\frac{5073129967862}{567947447479}a^{15}+\frac{4736952588582}{567947447479}a^{14}-\frac{4654172184820}{567947447479}a^{13}-\frac{7441782258858}{567947447479}a^{12}+\frac{15021861727732}{567947447479}a^{11}-\frac{6062759994278}{567947447479}a^{10}-\frac{7340362063133}{567947447479}a^{9}-\frac{160275627541}{567947447479}a^{8}+\frac{5521871973688}{567947447479}a^{7}-\frac{3626069384330}{567947447479}a^{6}+\frac{2430687681321}{567947447479}a^{5}+\frac{7966630250872}{567947447479}a^{4}+\frac{5302683238336}{567947447479}a^{3}+\frac{2638416515570}{567947447479}a^{2}+\frac{2051714839778}{567947447479}a+\frac{1070828353584}{567947447479}$, $a$, $\frac{105511272117}{567947447479}a^{19}-\frac{700060620917}{567947447479}a^{18}+\frac{2274242579151}{567947447479}a^{17}-\frac{4491201545370}{567947447479}a^{16}+\frac{5441443801948}{567947447479}a^{15}-\frac{3237609605119}{567947447479}a^{14}-\frac{243491085354}{567947447479}a^{13}+\frac{932030175769}{567947447479}a^{12}+\frac{2430514278548}{567947447479}a^{11}-\frac{5314227200785}{567947447479}a^{10}+\frac{3757355697435}{567947447479}a^{9}-\frac{1547783971598}{567947447479}a^{8}+\frac{682809465574}{567947447479}a^{7}-\frac{1289296337021}{567947447479}a^{6}+\frac{799137447853}{567947447479}a^{5}+\frac{1099272455264}{567947447479}a^{4}+\frac{988252572478}{567947447479}a^{3}+\frac{308046095328}{567947447479}a^{2}+\frac{381188507360}{567947447479}a+\frac{362654778810}{567947447479}$, $\frac{860131700202}{567947447479}a^{19}-\frac{4907115925957}{567947447479}a^{18}+\frac{12763738213433}{567947447479}a^{17}-\frac{17486434325192}{567947447479}a^{16}+\frac{7846858718881}{567947447479}a^{15}+\frac{12540269639065}{567947447479}a^{14}-\frac{13584944156328}{567947447479}a^{13}-\frac{12393337854399}{567947447479}a^{12}+\frac{30670497307325}{567947447479}a^{11}-\frac{13062329544587}{567947447479}a^{10}-\frac{16589309780021}{567947447479}a^{9}+\frac{5552387716280}{567947447479}a^{8}+\frac{9599621250822}{567947447479}a^{7}-\frac{5765487235215}{567947447479}a^{6}+\frac{3197111621079}{567947447479}a^{5}+\frac{12421469077706}{567947447479}a^{4}+\frac{7416722368587}{567947447479}a^{3}+\frac{894351039685}{567947447479}a^{2}+\frac{1173428128661}{567947447479}a+\frac{1250239218663}{567947447479}$, $\frac{28088777574}{567947447479}a^{19}-\frac{24382680215}{567947447479}a^{18}-\frac{427679203109}{567947447479}a^{17}+\frac{1825023181737}{567947447479}a^{16}-\frac{3509679186242}{567947447479}a^{15}+\frac{3199964570629}{567947447479}a^{14}+\frac{269403529900}{567947447479}a^{13}-\frac{2516812877846}{567947447479}a^{12}-\frac{281460870403}{567947447479}a^{11}+\frac{4138457426655}{567947447479}a^{10}-\frac{3536555395580}{567947447479}a^{9}-\frac{619734376528}{567947447479}a^{8}+\frac{19741141103}{567947447479}a^{7}+\frac{1961339118396}{567947447479}a^{6}-\frac{50384003732}{567947447479}a^{5}+\frac{478416946721}{567947447479}a^{4}+\frac{1346041852158}{567947447479}a^{3}+\frac{326474834375}{567947447479}a^{2}-\frac{1064497393267}{567947447479}a-\frac{182976152001}{567947447479}$, $\frac{269456357483}{567947447479}a^{19}-\frac{2063267299240}{567947447479}a^{18}+\frac{7127506917069}{567947447479}a^{17}-\frac{14089457937422}{567947447479}a^{16}+\frac{15569482286613}{567947447479}a^{15}-\frac{5091856461196}{567947447479}a^{14}-\frac{7939312548219}{567947447479}a^{13}+\frac{3928029004312}{567947447479}a^{12}+\frac{14567038421767}{567947447479}a^{11}-\frac{21781042462054}{567947447479}a^{10}+\frac{6887744750984}{567947447479}a^{9}+\frac{5342023633679}{567947447479}a^{8}+\frac{2748555486474}{567947447479}a^{7}-\frac{7867297966506}{567947447479}a^{6}+\frac{4111815515067}{567947447479}a^{5}+\frac{1315291911838}{567947447479}a^{4}-\frac{3816330034824}{567947447479}a^{3}-\frac{3085772263482}{567947447479}a^{2}+\frac{535158346358}{567947447479}a-\frac{396857135885}{567947447479}$
| 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: | \( 166.624941197 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 166.624941197 \cdot 1}{2\cdot\sqrt{2199140361717170013184}}\cr\approx \mathstrut & 0.170365723673 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 11*x^18 - 11*x^17 - 2*x^16 + 16*x^15 - 2*x^14 - 24*x^13 + 22*x^12 + 8*x^11 - 22*x^10 - 13*x^9 + 14*x^8 + 3*x^7 + x^6 + 15*x^5 + 20*x^4 + 9*x^3 + 3*x^2 + 2*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^20 - 5*x^19 + 11*x^18 - 11*x^17 - 2*x^16 + 16*x^15 - 2*x^14 - 24*x^13 + 22*x^12 + 8*x^11 - 22*x^10 - 13*x^9 + 14*x^8 + 3*x^7 + x^6 + 15*x^5 + 20*x^4 + 9*x^3 + 3*x^2 + 2*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^20 - 5*x^19 + 11*x^18 - 11*x^17 - 2*x^16 + 16*x^15 - 2*x^14 - 24*x^13 + 22*x^12 + 8*x^11 - 22*x^10 - 13*x^9 + 14*x^8 + 3*x^7 + x^6 + 15*x^5 + 20*x^4 + 9*x^3 + 3*x^2 + 2*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^20 - 5*x^19 + 11*x^18 - 11*x^17 - 2*x^16 + 16*x^15 - 2*x^14 - 24*x^13 + 22*x^12 + 8*x^11 - 22*x^10 - 13*x^9 + 14*x^8 + 3*x^7 + x^6 + 15*x^5 + 20*x^4 + 9*x^3 + 3*x^2 + 2*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
$C_2^9.C_2^5.S_5$ (as 20T994):
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 1966080 |
| The 280 conjugacy class representatives for $C_2^9.C_2^5.S_5$ |
| Character table for $C_2^9.C_2^5.S_5$ |
Intermediate fields
| 5.1.3017.1, 10.0.209352647.1 |
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 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 20.2.97909553495581830152192.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 2.10.10.3 | $x^{10} + 10 x^{9} + 74 x^{8} + 320 x^{7} + 1104 x^{6} + 2752 x^{5} + 6176 x^{4} + 12096 x^{3} + 17712 x^{2} + 15968 x + 8416$ | $2$ | $5$ | $10$ | $C_2^4 : C_5$ | $[2, 2, 2, 2]^{5}$ | |
|
\(7\)
| 7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 7.8.6.2 | $x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2660 x^{4} + 3408 x^{3} + 3312 x^{2} + 5184 x + 6304$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
|
\(23\)
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(431\)
| $\Q_{431}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{431}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $4$ | $2$ | $2$ | $2$ | ||||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |