Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 36*x^13 - 27*x^12 + 459*x^11 + 540*x^10 - 2560*x^9 - 3357*x^8 + 7287*x^7 + 8848*x^6 - 11637*x^5 - 9867*x^4 + 10811*x^3 + 3105*x^2 - 4623*x + 989)
gp: K = bnfinit(y^15 - 36*y^13 - 27*y^12 + 459*y^11 + 540*y^10 - 2560*y^9 - 3357*y^8 + 7287*y^7 + 8848*y^6 - 11637*y^5 - 9867*y^4 + 10811*y^3 + 3105*y^2 - 4623*y + 989, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 36*x^13 - 27*x^12 + 459*x^11 + 540*x^10 - 2560*x^9 - 3357*x^8 + 7287*x^7 + 8848*x^6 - 11637*x^5 - 9867*x^4 + 10811*x^3 + 3105*x^2 - 4623*x + 989);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 36*x^13 - 27*x^12 + 459*x^11 + 540*x^10 - 2560*x^9 - 3357*x^8 + 7287*x^7 + 8848*x^6 - 11637*x^5 - 9867*x^4 + 10811*x^3 + 3105*x^2 - 4623*x + 989)
\( x^{15} - 36 x^{13} - 27 x^{12} + 459 x^{11} + 540 x^{10} - 2560 x^{9} - 3357 x^{8} + 7287 x^{7} + \cdots + 989 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[15, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(262047125362808326177593\)
\(\medspace = 3^{15}\cdot 11^{13}\cdot 23^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(36.41\) | 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: | $3^{241/162}11^{9/10}23^{2/3}\approx 358.80911756371285$ | ||
| Ramified primes: |
\(3\), \(11\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{33}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{296023132241}a^{14}+\frac{14541677572}{296023132241}a^{13}-\frac{24259727745}{296023132241}a^{12}+\frac{33251684599}{296023132241}a^{11}-\frac{24718356183}{296023132241}a^{10}-\frac{25729467319}{296023132241}a^{9}+\frac{134577557377}{296023132241}a^{8}-\frac{138548707361}{296023132241}a^{7}+\frac{87814826151}{296023132241}a^{6}-\frac{142175864933}{296023132241}a^{5}+\frac{64816657097}{296023132241}a^{4}+\frac{19852719515}{296023132241}a^{3}-\frac{1183406442}{12870570967}a^{2}+\frac{5500827734}{12870570967}a+\frac{659648998}{12870570967}$
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: |
$\frac{138644595603}{296023132241}a^{14}-\frac{302933891682}{296023132241}a^{13}-\frac{4314634566851}{296023132241}a^{12}+\frac{5667980887404}{296023132241}a^{11}+\frac{50766244844559}{296023132241}a^{10}-\frac{35931193079618}{296023132241}a^{9}-\frac{270635463034812}{296023132241}a^{8}+\frac{127328162405229}{296023132241}a^{7}+\frac{702639063514558}{296023132241}a^{6}-\frac{319681392432462}{296023132241}a^{5}-\frac{843342020741973}{296023132241}a^{4}+\frac{495611375425106}{296023132241}a^{3}+\frac{14515856683389}{12870570967}a^{2}-\frac{13186958902059}{12870570967}a+\frac{2538575100934}{12870570967}$, $\frac{126966931911}{296023132241}a^{14}-\frac{282588889176}{296023132241}a^{13}-\frac{3934788619063}{296023132241}a^{12}+\frac{5309001204831}{296023132241}a^{11}+\frac{46243210726110}{296023132241}a^{10}-\frac{33907367665694}{296023132241}a^{9}-\frac{246876220389564}{296023132241}a^{8}+\frac{119617051483278}{296023132241}a^{7}+\frac{642967659858278}{296023132241}a^{6}-\frac{293890918141062}{296023132241}a^{5}-\frac{776052442689432}{296023132241}a^{4}+\frac{447010288763323}{296023132241}a^{3}+\frac{13583837249208}{12870570967}a^{2}-\frac{11799551663052}{12870570967}a+\frac{2160540300379}{12870570967}$, $\frac{94360753035}{296023132241}a^{14}-\frac{202179545766}{296023132241}a^{13}-\frac{2922939384418}{296023132241}a^{12}+\frac{3654905982444}{296023132241}a^{11}+\frac{34175894593668}{296023132241}a^{10}-\frac{21547774713862}{296023132241}a^{9}-\frac{180193146288111}{296023132241}a^{8}+\frac{69724624604973}{296023132241}a^{7}+\frac{461451641319845}{296023132241}a^{6}-\frac{170516515230972}{296023132241}a^{5}-\frac{547445618527182}{296023132241}a^{4}+\frac{275504368373396}{296023132241}a^{3}+\frac{9549857255652}{12870570967}a^{2}-\frac{7634160500463}{12870570967}a+\frac{1367533553155}{12870570967}$, $\frac{209650021254}{296023132241}a^{14}-\frac{464423432436}{296023132241}a^{13}-\frac{6477882055693}{296023132241}a^{12}+\frac{8604927504702}{296023132241}a^{11}+\frac{75896071201329}{296023132241}a^{10}-\frac{53431316965632}{296023132241}a^{9}-\frac{403310124032427}{296023132241}a^{8}+\frac{181630565166300}{296023132241}a^{7}+\frac{10\!\cdots\!43}{296023132241}a^{6}-\frac{438616959080634}{296023132241}a^{5}-\frac{12\!\cdots\!73}{296023132241}a^{4}+\frac{673913570474936}{296023132241}a^{3}+\frac{22201675070679}{12870570967}a^{2}-\frac{18046304924508}{12870570967}a+\frac{3150039052979}{12870570967}$, $\frac{9258399337}{12870570967}a^{14}-\frac{22857407975}{12870570967}a^{13}-\frac{281934683086}{12870570967}a^{12}+\frac{454872457854}{12870570967}a^{11}+\frac{3287027781011}{12870570967}a^{10}-\frac{3247552119360}{12870570967}a^{9}-\frac{17570536263167}{12870570967}a^{8}+\frac{12712613294794}{12870570967}a^{7}+\frac{45909151939509}{12870570967}a^{6}-\frac{31399028716490}{12870570967}a^{5}-\frac{54988851116098}{12870570967}a^{4}+\frac{44315807668969}{12870570967}a^{3}+\frac{20165230632917}{12870570967}a^{2}-\frac{25026103590686}{12870570967}a+\frac{5548568546788}{12870570967}$, $\frac{216817998305}{296023132241}a^{14}-\frac{419975647147}{296023132241}a^{13}-\frac{6813555123145}{296023132241}a^{12}+\frac{7010668584643}{296023132241}a^{11}+\frac{80301695804817}{296023132241}a^{10}-\frac{33081844479212}{296023132241}a^{9}-\frac{424255708847159}{296023132241}a^{8}+\frac{71429886884449}{296023132241}a^{7}+\frac{10\!\cdots\!36}{296023132241}a^{6}-\frac{157480532574288}{296023132241}a^{5}-\frac{13\!\cdots\!24}{296023132241}a^{4}+\frac{342415665410800}{296023132241}a^{3}+\frac{25124834696642}{12870570967}a^{2}-\frac{11663387600426}{12870570967}a+\frac{1000026613029}{12870570967}$, $\frac{86583784396}{296023132241}a^{14}-\frac{216628519303}{296023132241}a^{13}-\frac{2606999226707}{296023132241}a^{12}+\frac{4244128388091}{296023132241}a^{11}+\frac{30121819086314}{296023132241}a^{10}-\frac{29541813326881}{296023132241}a^{9}-\frac{159339951214485}{296023132241}a^{8}+\frac{111773649564190}{296023132241}a^{7}+\frac{410936301508517}{296023132241}a^{6}-\frac{268658790483993}{296023132241}a^{5}-\frac{484275858522779}{296023132241}a^{4}+\frac{375500674712643}{296023132241}a^{3}+\frac{7537899332421}{12870570967}a^{2}-\frac{9210538253834}{12870570967}a+\frac{2125208116493}{12870570967}$, $\frac{258400775943}{296023132241}a^{14}-\frac{629902610675}{296023132241}a^{13}-\frac{7841341993645}{296023132241}a^{12}+\frac{12268540645589}{296023132241}a^{11}+\frac{91058420487242}{296023132241}a^{10}-\frac{84407046157749}{296023132241}a^{9}-\frac{483561175963077}{296023132241}a^{8}+\frac{317539558699161}{296023132241}a^{7}+\frac{12\!\cdots\!99}{296023132241}a^{6}-\frac{769374010586864}{296023132241}a^{5}-\frac{14\!\cdots\!70}{296023132241}a^{4}+\frac{10\!\cdots\!69}{296023132241}a^{3}+\frac{24554548850266}{12870570967}a^{2}-\frac{27075733439312}{12870570967}a+\frac{5687120048340}{12870570967}$, $\frac{67536591552}{296023132241}a^{14}-\frac{116617935523}{296023132241}a^{13}-\frac{2180492849088}{296023132241}a^{12}+\frac{1875029948449}{296023132241}a^{11}+\frac{26094432119467}{296023132241}a^{10}-\frac{7666582414743}{296023132241}a^{9}-\frac{139211007190695}{296023132241}a^{8}+\frac{12442626519438}{296023132241}a^{7}+\frac{359859206561562}{296023132241}a^{6}-\frac{34773615225853}{296023132241}a^{5}-\frac{435999664948082}{296023132241}a^{4}+\frac{103728827627710}{296023132241}a^{3}+\frac{8473476285260}{12870570967}a^{2}-\frac{3841070404702}{12870570967}a+\frac{306444669647}{12870570967}$, $\frac{69185414883}{296023132241}a^{14}-\frac{216495524922}{296023132241}a^{13}-\frac{2027506564770}{296023132241}a^{12}+\frac{4865529233031}{296023132241}a^{11}+\frac{23326148005958}{296023132241}a^{10}-\frac{41776388131532}{296023132241}a^{9}-\frac{126855854093981}{296023132241}a^{8}+\frac{188322848837379}{296023132241}a^{7}+\frac{340470029814745}{296023132241}a^{6}-\frac{476111926570574}{296023132241}a^{5}-\frac{407229967856111}{296023132241}a^{4}+\frac{626358322202964}{296023132241}a^{3}+\frac{4764033684313}{12870570967}a^{2}-\frac{14140738667860}{12870570967}a+\frac{3786599136800}{12870570967}$, $\frac{97854921390}{296023132241}a^{14}-\frac{228985362511}{296023132241}a^{13}-\frac{3015809921656}{296023132241}a^{12}+\frac{4466379545610}{296023132241}a^{11}+\frac{35367041278537}{296023132241}a^{10}-\frac{30684341063135}{296023132241}a^{9}-\frac{189207271371030}{296023132241}a^{8}+\frac{116769453736285}{296023132241}a^{7}+\frac{493944156131024}{296023132241}a^{6}-\frac{291662492809287}{296023132241}a^{5}-\frac{593259625539261}{296023132241}a^{4}+\frac{427555775428822}{296023132241}a^{3}+\frac{9778101887690}{12870570967}a^{2}-\frac{10877290057715}{12870570967}a+\frac{2329888755010}{12870570967}$, $\frac{52669812506}{296023132241}a^{14}-\frac{105107361396}{296023132241}a^{13}-\frac{1639705298262}{296023132241}a^{12}+\frac{1782714615201}{296023132241}a^{11}+\frac{19143587487485}{296023132241}a^{10}-\frac{8991383785590}{296023132241}a^{9}-\frac{100013712210031}{296023132241}a^{8}+\frac{23592109171424}{296023132241}a^{7}+\frac{253905873732884}{296023132241}a^{6}-\frac{57308878846256}{296023132241}a^{5}-\frac{302518997511356}{296023132241}a^{4}+\frac{107767938592042}{296023132241}a^{3}+\frac{5630516150627}{12870570967}a^{2}-\frac{3333040088071}{12870570967}a+\frac{437731416563}{12870570967}$, $\frac{106043035937}{296023132241}a^{14}-\frac{361050259944}{296023132241}a^{13}-\frac{3042956821901}{296023132241}a^{12}+\frac{8333692704475}{296023132241}a^{11}+\frac{34654900593664}{296023132241}a^{10}-\frac{73972950032138}{296023132241}a^{9}-\frac{188921594967596}{296023132241}a^{8}+\frac{338980265059662}{296023132241}a^{7}+\frac{509531554317474}{296023132241}a^{6}-\frac{851831390122013}{296023132241}a^{5}-\frac{603675960919969}{296023132241}a^{4}+\frac{10\!\cdots\!72}{296023132241}a^{3}+\frac{5821630724166}{12870570967}a^{2}-\frac{24107590494031}{12870570967}a+\frac{6817627212906}{12870570967}$, $\frac{499898401}{296023132241}a^{14}+\frac{49811301478}{296023132241}a^{13}-\frac{96762798164}{296023132241}a^{12}-\frac{1613648227991}{296023132241}a^{11}+\frac{1308279131975}{296023132241}a^{10}+\frac{19215339971097}{296023132241}a^{9}-\frac{2911515915715}{296023132241}a^{8}-\frac{100247712748961}{296023132241}a^{7}-\frac{9079154466804}{296023132241}a^{6}+\frac{249151508257959}{296023132241}a^{5}+\frac{23621845153582}{296023132241}a^{4}-\frac{285978744990456}{296023132241}a^{3}+\frac{798166543547}{12870570967}a^{2}+\frac{5349749945050}{12870570967}a-\frac{1684644544994}{12870570967}$
| 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: | \( 6722708.12403 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 6722708.12403 \cdot 1}{2\cdot\sqrt{262047125362808326177593}}\cr\approx \mathstrut & 0.215166420740 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 36*x^13 - 27*x^12 + 459*x^11 + 540*x^10 - 2560*x^9 - 3357*x^8 + 7287*x^7 + 8848*x^6 - 11637*x^5 - 9867*x^4 + 10811*x^3 + 3105*x^2 - 4623*x + 989)
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^15 - 36*x^13 - 27*x^12 + 459*x^11 + 540*x^10 - 2560*x^9 - 3357*x^8 + 7287*x^7 + 8848*x^6 - 11637*x^5 - 9867*x^4 + 10811*x^3 + 3105*x^2 - 4623*x + 989, 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^15 - 36*x^13 - 27*x^12 + 459*x^11 + 540*x^10 - 2560*x^9 - 3357*x^8 + 7287*x^7 + 8848*x^6 - 11637*x^5 - 9867*x^4 + 10811*x^3 + 3105*x^2 - 4623*x + 989);
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^15 - 36*x^13 - 27*x^12 + 459*x^11 + 540*x^10 - 2560*x^9 - 3357*x^8 + 7287*x^7 + 8848*x^6 - 11637*x^5 - 9867*x^4 + 10811*x^3 + 3105*x^2 - 4623*x + 989);
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
$C_7^3:C_6$ (as 15T44):
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 2430 |
| The 39 conjugacy class representatives for $C_7^3:C_6$ |
| Character table for $C_7^3:C_6$ is not computed |
Intermediate fields
| \(\Q(\zeta_{11})^+\) |
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 15 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 45 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $15$ | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | R | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ | $15$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | R | $15$ | $15$ | $15$ | ${\href{/padicField/41.5.0.1}{5} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.15.15.29 | $x^{15} - 24 x^{14} + 387 x^{13} - 975 x^{12} + 29016 x^{11} + 513 x^{10} + 106173 x^{9} + 431487 x^{8} + 296460 x^{7} - 2322 x^{6} + 22437 x^{5} + 49086 x^{4} + 45603 x^{3} + 6318 x^{2} - 1458 x + 243$ | $3$ | $5$ | $15$ | 15T33 | $[3/2, 3/2, 3/2, 3/2]_{2}^{5}$ |
|
\(11\)
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
|
\(23\)
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.3.2.1 | $x^{3} + 23$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |