LMFDB - Number field 24.0.133084684332123489494188901166116961.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 3*x^22 - 8*x^21 + 8*x^20 - 24*x^19 + 37*x^18 - 120*x^17 + 194*x^16 - 329*x^15 + 744*x^14 - 904*x^13 + 1633*x^12 - 2712*x^11 + 6696*x^10 - 8883*x^9 + 15714*x^8 - 29160*x^7 + 26973*x^6 - 52488*x^5 + 52488*x^4 - 157464*x^3 + 177147*x^2 - 177147*x + 531441)

gp: K = bnfinit(y^24 - y^23 + 3*y^22 - 8*y^21 + 8*y^20 - 24*y^19 + 37*y^18 - 120*y^17 + 194*y^16 - 329*y^15 + 744*y^14 - 904*y^13 + 1633*y^12 - 2712*y^11 + 6696*y^10 - 8883*y^9 + 15714*y^8 - 29160*y^7 + 26973*y^6 - 52488*y^5 + 52488*y^4 - 157464*y^3 + 177147*y^2 - 177147*y + 531441, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 + 3*x^22 - 8*x^21 + 8*x^20 - 24*x^19 + 37*x^18 - 120*x^17 + 194*x^16 - 329*x^15 + 744*x^14 - 904*x^13 + 1633*x^12 - 2712*x^11 + 6696*x^10 - 8883*x^9 + 15714*x^8 - 29160*x^7 + 26973*x^6 - 52488*x^5 + 52488*x^4 - 157464*x^3 + 177147*x^2 - 177147*x + 531441);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - x^23 + 3*x^22 - 8*x^21 + 8*x^20 - 24*x^19 + 37*x^18 - 120*x^17 + 194*x^16 - 329*x^15 + 744*x^14 - 904*x^13 + 1633*x^12 - 2712*x^11 + 6696*x^10 - 8883*x^9 + 15714*x^8 - 29160*x^7 + 26973*x^6 - 52488*x^5 + 52488*x^4 - 157464*x^3 + 177147*x^2 - 177147*x + 531441)

$ x^{24} - x^{23} + 3 x^{22} - 8 x^{21} + 8 x^{20} - 24 x^{19} + 37 x^{18} - 120 x^{17} + 194 x^{16} + \cdots + 531441 $ x^24 - x^23 + 3*x^22 - 8*x^21 + 8*x^20 - 24*x^19 + 37*x^18 - 120*x^17 + 194*x^16 - 329*x^15 + 744*x^14 - 904*x^13 + 1633*x^12 - 2712*x^11 + 6696*x^10 - 8883*x^9 + 15714*x^8 - 29160*x^7 + 26973*x^6 - 52488*x^5 + 52488*x^4 - 157464*x^3 + 177147*x^2 - 177147*x + 531441

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$24$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 12]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$133084684332123489494188901166116961$ 133084684332123489494188901166116961 $\medspace = 3^{12}\cdot 7^{20}\cdot 11^{12}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$29.07$	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:	$3^{1/2}7^{5/6}11^{1/2}\approx 29.074036854473896$
Ramified primes:	$3$, $7$, $11$ 3, 7, 11	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$24$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$231=3\cdot 7\cdot 11$
Dirichlet character group:	$\lbrace$$\chi_{231}(1,·)$, $\chi_{231}(131,·)$, $\chi_{231}(197,·)$, $\chi_{231}(100,·)$, $\chi_{231}(65,·)$, $\chi_{231}(10,·)$, $\chi_{231}(76,·)$, $\chi_{231}(142,·)$, $\chi_{231}(208,·)$, $\chi_{231}(67,·)$, $\chi_{231}(23,·)$, $\chi_{231}(89,·)$, $\chi_{231}(155,·)$, $\chi_{231}(221,·)$, $\chi_{231}(199,·)$, $\chi_{231}(32,·)$, $\chi_{231}(34,·)$, $\chi_{231}(164,·)$, $\chi_{231}(230,·)$, $\chi_{231}(166,·)$, $\chi_{231}(43,·)$, $\chi_{231}(109,·)$, $\chi_{231}(122,·)$, $\chi_{231}(188,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{2048}$

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{117}a^{14}-\frac{1}{9}a^{13}-\frac{1}{3}a^{12}-\frac{2}{9}a^{11}-\frac{1}{9}a^{10}-\frac{1}{3}a^{9}-\frac{2}{9}a^{8}+\frac{10}{39}a^{7}-\frac{1}{9}a^{6}+\frac{4}{9}a^{5}+\frac{1}{3}a^{4}-\frac{1}{9}a^{3}+\frac{4}{9}a^{2}+\frac{1}{3}a-\frac{4}{13}$, $\frac{1}{351}a^{15}-\frac{1}{351}a^{14}+\frac{1}{9}a^{13}+\frac{7}{27}a^{12}+\frac{11}{27}a^{11}+\frac{1}{9}a^{10}-\frac{2}{27}a^{9}+\frac{23}{117}a^{8}-\frac{121}{351}a^{7}+\frac{10}{27}a^{6}-\frac{4}{9}a^{5}+\frac{8}{27}a^{4}+\frac{1}{27}a^{3}-\frac{4}{9}a^{2}-\frac{17}{39}a-\frac{3}{13}$, $\frac{1}{1053}a^{16}-\frac{1}{1053}a^{15}+\frac{1}{351}a^{14}-\frac{11}{81}a^{13}+\frac{11}{81}a^{12}-\frac{11}{27}a^{11}-\frac{20}{81}a^{10}+\frac{140}{351}a^{9}+\frac{113}{1053}a^{8}+\frac{103}{1053}a^{7}-\frac{10}{27}a^{6}-\frac{28}{81}a^{5}+\frac{28}{81}a^{4}-\frac{1}{27}a^{3}+\frac{16}{39}a^{2}-\frac{16}{39}a+\frac{3}{13}$, $\frac{1}{3159}a^{17}-\frac{1}{3159}a^{16}+\frac{1}{1053}a^{15}-\frac{8}{3159}a^{14}+\frac{38}{243}a^{13}-\frac{38}{81}a^{12}+\frac{115}{243}a^{11}-\frac{445}{1053}a^{10}+\frac{1166}{3159}a^{9}-\frac{1301}{3159}a^{8}+\frac{167}{1053}a^{7}+\frac{80}{243}a^{6}-\frac{80}{243}a^{5}-\frac{1}{81}a^{4}-\frac{49}{117}a^{3}+\frac{49}{117}a^{2}-\frac{10}{39}a+\frac{6}{13}$, $\frac{1}{18954}a^{18}+\frac{1}{9477}a^{17}+\frac{1}{18954}a^{15}-\frac{8}{9477}a^{14}-\frac{227}{1458}a^{12}+\frac{526}{1053}a^{11}-\frac{2999}{9477}a^{10}-\frac{317}{1458}a^{9}+\frac{19}{39}a^{8}-\frac{4439}{9477}a^{7}-\frac{83}{1458}a^{6}-\frac{80}{1053}a^{4}+\frac{253}{702}a^{3}-\frac{2}{39}a-\frac{9}{26}$, $\frac{1}{133682562}a^{19}-\frac{638}{66841281}a^{18}+\frac{463}{7426809}a^{17}-\frac{10889}{133682562}a^{16}+\frac{15688}{66841281}a^{15}-\frac{3704}{7426809}a^{14}+\frac{1588741}{10283274}a^{13}+\frac{2135477}{7426809}a^{12}-\frac{33385187}{66841281}a^{11}-\frac{23423669}{133682562}a^{10}-\frac{570436}{7426809}a^{9}-\frac{24266540}{66841281}a^{8}+\frac{30788947}{133682562}a^{7}+\frac{266432}{571293}a^{6}-\frac{376343}{825201}a^{5}+\frac{630407}{1650402}a^{4}-\frac{78964}{275067}a^{3}+\frac{6782}{30563}a^{2}+\frac{9939}{61126}a-\frac{9026}{30563}$, $\frac{1}{401047686}a^{20}-\frac{1}{401047686}a^{19}-\frac{1747}{66841281}a^{18}-\frac{6857}{401047686}a^{17}-\frac{14113}{401047686}a^{16}+\frac{5309}{66841281}a^{15}+\frac{54829}{401047686}a^{14}+\frac{6067175}{133682562}a^{13}+\frac{68094349}{200523843}a^{12}+\frac{152359321}{401047686}a^{11}+\frac{24940817}{133682562}a^{10}+\frac{29282029}{200523843}a^{9}+\frac{12495343}{401047686}a^{8}+\frac{37707425}{133682562}a^{7}-\frac{10934744}{22280427}a^{6}+\frac{2757635}{14853618}a^{5}-\frac{920833}{4951206}a^{4}+\frac{364441}{825201}a^{3}+\frac{25547}{550134}a^{2}-\frac{16877}{61126}a-\frac{8622}{30563}$, $\frac{1}{1203143058}a^{21}-\frac{1}{1203143058}a^{20}+\frac{1}{401047686}a^{19}+\frac{305}{46274733}a^{18}-\frac{103429}{1203143058}a^{17}+\frac{39763}{401047686}a^{16}-\frac{174982}{601571529}a^{15}-\frac{550165}{401047686}a^{14}+\frac{94073027}{1203143058}a^{13}-\frac{2424130}{601571529}a^{12}+\frac{1513691}{30849822}a^{11}+\frac{582544049}{1203143058}a^{10}-\frac{46722778}{601571529}a^{9}-\frac{26025637}{401047686}a^{8}-\frac{4645441}{44560854}a^{7}+\frac{3508952}{22280427}a^{6}-\frac{2444959}{14853618}a^{5}+\frac{61487}{126954}a^{4}-\frac{150613}{825201}a^{3}+\frac{125191}{550134}a^{2}+\frac{79927}{183378}a+\frac{3817}{61126}$, $\frac{1}{3609429174}a^{22}-\frac{1}{3609429174}a^{21}+\frac{1}{1203143058}a^{20}-\frac{4}{1804714587}a^{19}-\frac{33692}{1804714587}a^{18}-\frac{175913}{1203143058}a^{17}+\frac{196484}{1804714587}a^{16}-\frac{620696}{601571529}a^{15}+\frac{6699785}{3609429174}a^{14}-\frac{51563428}{1804714587}a^{13}-\frac{71855759}{601571529}a^{12}+\frac{88239443}{3609429174}a^{11}-\frac{183324175}{1804714587}a^{10}-\frac{140792762}{601571529}a^{9}-\frac{979177}{133682562}a^{8}-\frac{25134913}{66841281}a^{7}-\frac{2781121}{22280427}a^{6}-\frac{62875}{550134}a^{5}-\frac{643358}{2475603}a^{4}-\frac{101704}{275067}a^{3}+\frac{173989}{550134}a^{2}+\frac{67975}{183378}a-\frac{17727}{61126}$, $\frac{1}{10828287522}a^{23}-\frac{1}{10828287522}a^{22}+\frac{1}{3609429174}a^{21}-\frac{4}{5414143761}a^{20}+\frac{4}{5414143761}a^{19}-\frac{42448}{1804714587}a^{18}+\frac{39545}{416472597}a^{17}-\frac{669242}{1804714587}a^{16}+\frac{3417892}{5414143761}a^{15}-\frac{7829503}{5414143761}a^{14}-\frac{133470299}{1804714587}a^{13}+\frac{2474930734}{5414143761}a^{12}-\frac{1780524457}{5414143761}a^{11}+\frac{2108956}{138824199}a^{10}+\frac{55081483}{200523843}a^{9}-\frac{74558275}{200523843}a^{8}-\frac{11001596}{66841281}a^{7}+\frac{3397267}{22280427}a^{6}-\frac{2227603}{7426809}a^{5}+\frac{447100}{2475603}a^{4}+\frac{1373}{7053}a^{3}+\frac{128431}{550134}a^{2}+\frac{57497}{183378}a-\frac{3427}{61126}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, 1/3*a^13 - 1/3*a^12 + 1/3*a^10 - 1/3*a^9 + 1/3*a^7 - 1/3*a^5 + 1/3*a^4 - 1/3*a^2 + 1/3*a, 1/117*a^14 - 1/9*a^13 - 1/3*a^12 - 2/9*a^11 - 1/9*a^10 - 1/3*a^9 - 2/9*a^8 + 10/39*a^7 - 1/9*a^6 + 4/9*a^5 + 1/3*a^4 - 1/9*a^3 + 4/9*a^2 + 1/3*a - 4/13, 1/351*a^15 - 1/351*a^14 + 1/9*a^13 + 7/27*a^12 + 11/27*a^11 + 1/9*a^10 - 2/27*a^9 + 23/117*a^8 - 121/351*a^7 + 10/27*a^6 - 4/9*a^5 + 8/27*a^4 + 1/27*a^3 - 4/9*a^2 - 17/39*a - 3/13, 1/1053*a^16 - 1/1053*a^15 + 1/351*a^14 - 11/81*a^13 + 11/81*a^12 - 11/27*a^11 - 20/81*a^10 + 140/351*a^9 + 113/1053*a^8 + 103/1053*a^7 - 10/27*a^6 - 28/81*a^5 + 28/81*a^4 - 1/27*a^3 + 16/39*a^2 - 16/39*a + 3/13, 1/3159*a^17 - 1/3159*a^16 + 1/1053*a^15 - 8/3159*a^14 + 38/243*a^13 - 38/81*a^12 + 115/243*a^11 - 445/1053*a^10 + 1166/3159*a^9 - 1301/3159*a^8 + 167/1053*a^7 + 80/243*a^6 - 80/243*a^5 - 1/81*a^4 - 49/117*a^3 + 49/117*a^2 - 10/39*a + 6/13, 1/18954*a^18 + 1/9477*a^17 + 1/18954*a^15 - 8/9477*a^14 - 227/1458*a^12 + 526/1053*a^11 - 2999/9477*a^10 - 317/1458*a^9 + 19/39*a^8 - 4439/9477*a^7 - 83/1458*a^6 - 80/1053*a^4 + 253/702*a^3 - 2/39*a - 9/26, 1/133682562*a^19 - 638/66841281*a^18 + 463/7426809*a^17 - 10889/133682562*a^16 + 15688/66841281*a^15 - 3704/7426809*a^14 + 1588741/10283274*a^13 + 2135477/7426809*a^12 - 33385187/66841281*a^11 - 23423669/133682562*a^10 - 570436/7426809*a^9 - 24266540/66841281*a^8 + 30788947/133682562*a^7 + 266432/571293*a^6 - 376343/825201*a^5 + 630407/1650402*a^4 - 78964/275067*a^3 + 6782/30563*a^2 + 9939/61126*a - 9026/30563, 1/401047686*a^20 - 1/401047686*a^19 - 1747/66841281*a^18 - 6857/401047686*a^17 - 14113/401047686*a^16 + 5309/66841281*a^15 + 54829/401047686*a^14 + 6067175/133682562*a^13 + 68094349/200523843*a^12 + 152359321/401047686*a^11 + 24940817/133682562*a^10 + 29282029/200523843*a^9 + 12495343/401047686*a^8 + 37707425/133682562*a^7 - 10934744/22280427*a^6 + 2757635/14853618*a^5 - 920833/4951206*a^4 + 364441/825201*a^3 + 25547/550134*a^2 - 16877/61126*a - 8622/30563, 1/1203143058*a^21 - 1/1203143058*a^20 + 1/401047686*a^19 + 305/46274733*a^18 - 103429/1203143058*a^17 + 39763/401047686*a^16 - 174982/601571529*a^15 - 550165/401047686*a^14 + 94073027/1203143058*a^13 - 2424130/601571529*a^12 + 1513691/30849822*a^11 + 582544049/1203143058*a^10 - 46722778/601571529*a^9 - 26025637/401047686*a^8 - 4645441/44560854*a^7 + 3508952/22280427*a^6 - 2444959/14853618*a^5 + 61487/126954*a^4 - 150613/825201*a^3 + 125191/550134*a^2 + 79927/183378*a + 3817/61126, 1/3609429174*a^22 - 1/3609429174*a^21 + 1/1203143058*a^20 - 4/1804714587*a^19 - 33692/1804714587*a^18 - 175913/1203143058*a^17 + 196484/1804714587*a^16 - 620696/601571529*a^15 + 6699785/3609429174*a^14 - 51563428/1804714587*a^13 - 71855759/601571529*a^12 + 88239443/3609429174*a^11 - 183324175/1804714587*a^10 - 140792762/601571529*a^9 - 979177/133682562*a^8 - 25134913/66841281*a^7 - 2781121/22280427*a^6 - 62875/550134*a^5 - 643358/2475603*a^4 - 101704/275067*a^3 + 173989/550134*a^2 + 67975/183378*a - 17727/61126, 1/10828287522*a^23 - 1/10828287522*a^22 + 1/3609429174*a^21 - 4/5414143761*a^20 + 4/5414143761*a^19 - 42448/1804714587*a^18 + 39545/416472597*a^17 - 669242/1804714587*a^16 + 3417892/5414143761*a^15 - 7829503/5414143761*a^14 - 133470299/1804714587*a^13 + 2474930734/5414143761*a^12 - 1780524457/5414143761*a^11 + 2108956/138824199*a^10 + 55081483/200523843*a^9 - 74558275/200523843*a^8 - 11001596/66841281*a^7 + 3397267/22280427*a^6 - 2227603/7426809*a^5 + 447100/2475603*a^4 + 1373/7053*a^3 + 128431/550134*a^2 + 57497/183378*a - 3427/61126

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

$C_{9}$, which has order $9$ (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:	$11$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ \frac{2573}{3609429174} a^{23} - \frac{2573}{1203143058} a^{22} + \frac{10292}{1804714587} a^{21} - \frac{10292}{1804714587} a^{20} + \frac{10292}{601571529} a^{19} - \frac{95201}{3609429174} a^{18} + \frac{81499}{1804714587} a^{17} - \frac{249581}{1804714587} a^{16} + \frac{846517}{3609429174} a^{15} - \frac{319052}{601571529} a^{14} + \frac{1162996}{1804714587} a^{13} - \frac{4201709}{3609429174} a^{12} + \frac{1162996}{601571529} a^{11} - \frac{12848555}{1804714587} a^{10} + \frac{846517}{133682562} a^{9} - \frac{249581}{22280427} a^{8} + \frac{51460}{2475603} a^{7} - \frac{95201}{4951206} a^{6} + \frac{10292}{275067} a^{5} - \frac{10292}{275067} a^{4} + \frac{13051}{183378} a^{3} - \frac{7719}{61126} a^{2} + \frac{7719}{61126} a - \frac{23157}{61126} $ (2573)/(3609429174)a^(23) - (2573)/(1203143058)a^(22) + (10292)/(1804714587)a^(21) - (10292)/(1804714587)a^(20) + (10292)/(601571529)a^(19) - (95201)/(3609429174)a^(18) + (81499)/(1804714587)a^(17) - (249581)/(1804714587)a^(16) + (846517)/(3609429174)a^(15) - (319052)/(601571529)a^(14) + (1162996)/(1804714587)a^(13) - (4201709)/(3609429174)a^(12) + (1162996)/(601571529)a^(11) - (12848555)/(1804714587)a^(10) + (846517)/(133682562)a^(9) - (249581)/(22280427)a^(8) + (51460)/(2475603)a^(7) - (95201)/(4951206)a^(6) + (10292)/(275067)a^(5) - (10292)/(275067)a^(4) + (13051)/(183378)a^(3) - (7719)/(61126)a^(2) + (7719)/(61126)*a - (23157)/(61126) (order $42$)	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:	$\frac{5}{2475603}a^{23}+\frac{1}{2475603}a^{22}+\frac{304}{22280427}a^{20}-\frac{8}{2475603}a^{19}-\frac{1}{30563}a^{18}-\frac{64}{2475603}a^{17}-\frac{14}{91689}a^{16}+\frac{565}{2475603}a^{15}-\frac{400}{2475603}a^{14}-\frac{8456}{22280427}a^{13}-\frac{2333}{2475603}a^{12}+\frac{4928}{2475603}a^{11}-\frac{536}{825201}a^{10}+\frac{997}{91689}a^{9}-\frac{886}{91689}a^{8}+\frac{472}{30563}a^{7}+\frac{37000}{22280427}a^{6}+\frac{720}{30563}a^{5}-\frac{3240}{30563}a^{4}+\frac{91144}{825201}a^{3}-\frac{8991}{30563}a^{2}+\frac{10935}{30563}a+\frac{2187}{30563}$, $\frac{8}{2475603}a^{23}-\frac{8}{2475603}a^{22}+\frac{8}{825201}a^{21}+\frac{88}{22280427}a^{20}+\frac{64}{2475603}a^{19}-\frac{64}{825201}a^{18}+\frac{296}{2475603}a^{17}-\frac{320}{825201}a^{16}+\frac{1552}{2475603}a^{15}-\frac{2632}{2475603}a^{14}+\frac{15952}{22280427}a^{13}-\frac{7232}{2475603}a^{12}+\frac{13064}{2475603}a^{11}-\frac{7232}{825201}a^{10}+\frac{1984}{91689}a^{9}-\frac{2632}{91689}a^{8}+\frac{1552}{30563}a^{7}-\frac{691271}{22280427}a^{6}+\frac{2664}{30563}a^{5}-\frac{5184}{30563}a^{4}+\frac{5184}{30563}a^{3}-\frac{15552}{30563}a^{2}+\frac{17496}{30563}a-\frac{48059}{30563}$, $\frac{67}{1203143058}a^{23}+\frac{43}{4951206}a^{22}-\frac{8}{825201}a^{21}-\frac{88}{22280427}a^{20}-\frac{64}{2475603}a^{19}+\frac{64}{825201}a^{18}-\frac{296}{2475603}a^{17}+\frac{4562}{46274733}a^{16}-\frac{1552}{2475603}a^{15}+\frac{2632}{2475603}a^{14}-\frac{15952}{22280427}a^{13}+\frac{7232}{2475603}a^{12}-\frac{13064}{2475603}a^{11}+\frac{7232}{825201}a^{10}-\frac{286063}{46274733}a^{9}+\frac{2632}{91689}a^{8}-\frac{1552}{30563}a^{7}+\frac{691271}{22280427}a^{6}-\frac{2664}{30563}a^{5}+\frac{5184}{30563}a^{4}-\frac{5184}{30563}a^{3}-\frac{48329}{550134}a^{2}-\frac{9379}{14106}a+\frac{48059}{30563}$, $\frac{1}{183378}a^{22}-\frac{16951}{183378}a+1$, $\frac{6640}{5414143761}a^{23}+\frac{104995}{10828287522}a^{22}+\frac{6640}{5414143761}a^{20}-\frac{47920}{5414143761}a^{19}-\frac{232400}{5414143761}a^{17}-\frac{6640}{601571529}a^{16}-\frac{3323935}{5414143761}a^{15}+\frac{1679920}{5414143761}a^{14}-\frac{6640}{22280427}a^{13}+\frac{10531799}{5414143761}a^{12}-\frac{551120}{416472597}a^{11}+\frac{6640}{2475603}a^{10}-\frac{1062400}{601571529}a^{9}+\frac{1481206}{66841281}a^{8}-\frac{33200}{2475603}a^{7}+\frac{491360}{22280427}a^{6}-\frac{720}{30563}a^{5}+\frac{6640}{190431}a^{4}-\frac{106240}{825201}a^{3}-\frac{142945}{183378}a+\frac{13280}{30563}$, $\frac{17}{4951206}a^{23}+\frac{211}{30849822}a^{22}+\frac{83}{10283274}a^{21}-\frac{1}{4951206}a^{20}-\frac{35}{14853618}a^{19}+\frac{35}{4951206}a^{17}+\frac{1}{550134}a^{16}-\frac{75562}{200523843}a^{15}-\frac{67957}{133682562}a^{14}+\frac{3}{61126}a^{13}+\frac{359}{7426809}a^{12}+\frac{83}{380862}a^{11}-\frac{27}{61126}a^{10}+\frac{80}{275067}a^{9}+\frac{2468431}{401047686}a^{8}+\frac{5368703}{133682562}a^{7}-\frac{111}{30563}a^{6}+\frac{933841}{14853618}a^{5}-\frac{27}{4702}a^{4}+\frac{648}{30563}a^{3}-\frac{108640}{275067}a^{2}-\frac{10935}{30563}a-\frac{93931}{61126}$, $\frac{10643}{1804714587}a^{23}+\frac{28589}{1203143058}a^{22}-\frac{52171}{3609429174}a^{21}-\frac{13132}{1804714587}a^{20}-\frac{61621}{1203143058}a^{19}+\frac{7369}{138824199}a^{18}-\frac{574021}{1804714587}a^{17}-\frac{494303}{3609429174}a^{16}-\frac{2893127}{1804714587}a^{15}+\frac{985145}{601571529}a^{14}-\frac{5160275}{3609429174}a^{13}+\frac{8032369}{1804714587}a^{12}-\frac{259300}{46274733}a^{11}+\frac{45487693}{3609429174}a^{10}-\frac{3075547}{601571529}a^{9}+\frac{17385371}{200523843}a^{8}-\frac{8262355}{133682562}a^{7}+\frac{1081052}{22280427}a^{6}-\frac{47132}{275067}a^{5}+\frac{34595}{380862}a^{4}-\frac{540767}{825201}a^{3}-\frac{144044}{275067}a^{2}-\frac{226817}{91689}a+\frac{91467}{61126}$, $\frac{24899}{10828287522}a^{23}-\frac{94457}{10828287522}a^{22}+\frac{64076}{1804714587}a^{21}-\frac{138685}{10828287522}a^{20}+\frac{1087891}{10828287522}a^{19}-\frac{576341}{3609429174}a^{18}+\frac{1695461}{10828287522}a^{17}-\frac{2033887}{3609429174}a^{16}+\frac{8457451}{10828287522}a^{15}-\frac{31598857}{10828287522}a^{14}+\frac{5337491}{3609429174}a^{13}-\frac{56523107}{10828287522}a^{12}+\frac{110655323}{10828287522}a^{11}-\frac{48342037}{3609429174}a^{10}+\frac{8182861}{401047686}a^{9}-\frac{1959713}{44560854}a^{8}+\frac{4975481}{44560854}a^{7}-\frac{361141}{4951206}a^{6}+\frac{2127863}{14853618}a^{5}-\frac{879577}{4951206}a^{4}+\frac{63857}{183378}a^{3}-\frac{40913}{275067}a^{2}-\frac{308}{30563}a-\frac{113843}{61126}$, $\frac{50533}{5414143761}a^{23}-\frac{75856}{5414143761}a^{22}+\frac{12472}{601571529}a^{21}-\frac{510763}{10828287522}a^{20}+\frac{470984}{5414143761}a^{19}-\frac{261031}{1203143058}a^{18}+\frac{4864055}{10828287522}a^{17}-\frac{298498}{200523843}a^{16}+\frac{25799515}{10828287522}a^{15}-\frac{31892719}{10828287522}a^{14}+\frac{3327119}{601571529}a^{13}-\frac{110846225}{10828287522}a^{12}+\frac{175228637}{10828287522}a^{11}-\frac{15705307}{601571529}a^{10}+\frac{84216077}{1203143058}a^{9}-\frac{41496035}{401047686}a^{8}+\frac{2673080}{22280427}a^{7}-\frac{7255835}{44560854}a^{6}+\frac{4052545}{14853618}a^{5}-\frac{1225840}{2475603}a^{4}+\frac{909125}{1650402}a^{3}-\frac{72605}{42318}a^{2}+\frac{68479}{30563}a-\frac{42769}{61126}$, $\frac{47822}{5414143761}a^{23}-\frac{195571}{10828287522}a^{22}+\frac{40256}{1804714587}a^{21}-\frac{431963}{10828287522}a^{20}+\frac{468544}{5414143761}a^{19}-\frac{674903}{3609429174}a^{18}+\frac{1404757}{10828287522}a^{17}-\frac{1199045}{1804714587}a^{16}+\frac{21482105}{10828287522}a^{15}-\frac{28125599}{10828287522}a^{14}+\frac{6503944}{1804714587}a^{13}-\frac{81642577}{10828287522}a^{12}+\frac{142134637}{10828287522}a^{11}-\frac{19657337}{1804714587}a^{10}+\frac{40573889}{1203143058}a^{9}-\frac{10937405}{133682562}a^{8}+\frac{78160}{825201}a^{7}-\frac{5644045}{44560854}a^{6}+\frac{3457585}{14853618}a^{5}-\frac{288760}{825201}a^{4}+\frac{169573}{1650402}a^{3}-\frac{107527}{183378}a^{2}+\frac{97705}{61126}a-\frac{27579}{61126}$, $\frac{3787}{3609429174}a^{23}-\frac{3938}{601571529}a^{22}-\frac{31622}{1804714587}a^{21}-\frac{13679}{3609429174}a^{20}+\frac{17243}{1203143058}a^{19}-\frac{66637}{3609429174}a^{18}+\frac{336367}{3609429174}a^{17}+\frac{687889}{3609429174}a^{16}+\frac{471515}{3609429174}a^{15}+\frac{2196229}{1203143058}a^{14}+\frac{1450957}{3609429174}a^{13}-\frac{3115507}{3609429174}a^{12}+\frac{1365907}{1203143058}a^{11}-\frac{1996871}{3609429174}a^{10}-\frac{13613569}{1203143058}a^{9}-\frac{248549}{44560854}a^{8}-\frac{11873117}{133682562}a^{7}-\frac{48655}{4951206}a^{6}-\frac{544825}{14853618}a^{5}-\frac{11249}{550134}a^{4}-\frac{54803}{1650402}a^{3}+\frac{10393}{91689}a^{2}+\frac{22225}{183378}a+\frac{165747}{61126}$ 5/2475603a^23 + 1/2475603a^22 + 304/22280427a^20 - 8/2475603a^19 - 1/30563a^18 - 64/2475603a^17 - 14/91689a^16 + 565/2475603a^15 - 400/2475603a^14 - 8456/22280427a^13 - 2333/2475603a^12 + 4928/2475603a^11 - 536/825201a^10 + 997/91689a^9 - 886/91689a^8 + 472/30563a^7 + 37000/22280427a^6 + 720/30563a^5 - 3240/30563a^4 + 91144/825201a^3 - 8991/30563a^2 + 10935/30563a + 2187/30563, 8/2475603a^23 - 8/2475603a^22 + 8/825201a^21 + 88/22280427a^20 + 64/2475603a^19 - 64/825201a^18 + 296/2475603a^17 - 320/825201a^16 + 1552/2475603a^15 - 2632/2475603a^14 + 15952/22280427a^13 - 7232/2475603a^12 + 13064/2475603a^11 - 7232/825201a^10 + 1984/91689a^9 - 2632/91689a^8 + 1552/30563a^7 - 691271/22280427a^6 + 2664/30563a^5 - 5184/30563a^4 + 5184/30563a^3 - 15552/30563a^2 + 17496/30563a - 48059/30563, 67/1203143058a^23 + 43/4951206a^22 - 8/825201a^21 - 88/22280427a^20 - 64/2475603a^19 + 64/825201a^18 - 296/2475603a^17 + 4562/46274733a^16 - 1552/2475603a^15 + 2632/2475603a^14 - 15952/22280427a^13 + 7232/2475603a^12 - 13064/2475603a^11 + 7232/825201a^10 - 286063/46274733a^9 + 2632/91689a^8 - 1552/30563a^7 + 691271/22280427a^6 - 2664/30563a^5 + 5184/30563a^4 - 5184/30563a^3 - 48329/550134a^2 - 9379/14106a + 48059/30563, 1/183378a^22 - 16951/183378a + 1, 6640/5414143761a^23 + 104995/10828287522a^22 + 6640/5414143761a^20 - 47920/5414143761a^19 - 232400/5414143761a^17 - 6640/601571529a^16 - 3323935/5414143761a^15 + 1679920/5414143761a^14 - 6640/22280427a^13 + 10531799/5414143761a^12 - 551120/416472597a^11 + 6640/2475603a^10 - 1062400/601571529a^9 + 1481206/66841281a^8 - 33200/2475603a^7 + 491360/22280427a^6 - 720/30563a^5 + 6640/190431a^4 - 106240/825201a^3 - 142945/183378a + 13280/30563, 17/4951206a^23 + 211/30849822a^22 + 83/10283274a^21 - 1/4951206a^20 - 35/14853618a^19 + 35/4951206a^17 + 1/550134a^16 - 75562/200523843a^15 - 67957/133682562a^14 + 3/61126a^13 + 359/7426809a^12 + 83/380862a^11 - 27/61126a^10 + 80/275067a^9 + 2468431/401047686a^8 + 5368703/133682562a^7 - 111/30563a^6 + 933841/14853618a^5 - 27/4702a^4 + 648/30563a^3 - 108640/275067a^2 - 10935/30563a - 93931/61126, 10643/1804714587a^23 + 28589/1203143058a^22 - 52171/3609429174a^21 - 13132/1804714587a^20 - 61621/1203143058a^19 + 7369/138824199a^18 - 574021/1804714587a^17 - 494303/3609429174a^16 - 2893127/1804714587a^15 + 985145/601571529a^14 - 5160275/3609429174a^13 + 8032369/1804714587a^12 - 259300/46274733a^11 + 45487693/3609429174a^10 - 3075547/601571529a^9 + 17385371/200523843a^8 - 8262355/133682562a^7 + 1081052/22280427a^6 - 47132/275067a^5 + 34595/380862a^4 - 540767/825201a^3 - 144044/275067a^2 - 226817/91689a + 91467/61126, 24899/10828287522a^23 - 94457/10828287522a^22 + 64076/1804714587a^21 - 138685/10828287522a^20 + 1087891/10828287522a^19 - 576341/3609429174a^18 + 1695461/10828287522a^17 - 2033887/3609429174a^16 + 8457451/10828287522a^15 - 31598857/10828287522a^14 + 5337491/3609429174a^13 - 56523107/10828287522a^12 + 110655323/10828287522a^11 - 48342037/3609429174a^10 + 8182861/401047686a^9 - 1959713/44560854a^8 + 4975481/44560854a^7 - 361141/4951206a^6 + 2127863/14853618a^5 - 879577/4951206a^4 + 63857/183378a^3 - 40913/275067a^2 - 308/30563a - 113843/61126, 50533/5414143761a^23 - 75856/5414143761a^22 + 12472/601571529a^21 - 510763/10828287522a^20 + 470984/5414143761a^19 - 261031/1203143058a^18 + 4864055/10828287522a^17 - 298498/200523843a^16 + 25799515/10828287522a^15 - 31892719/10828287522a^14 + 3327119/601571529a^13 - 110846225/10828287522a^12 + 175228637/10828287522a^11 - 15705307/601571529a^10 + 84216077/1203143058a^9 - 41496035/401047686a^8 + 2673080/22280427a^7 - 7255835/44560854a^6 + 4052545/14853618a^5 - 1225840/2475603a^4 + 909125/1650402a^3 - 72605/42318a^2 + 68479/30563a - 42769/61126, 47822/5414143761a^23 - 195571/10828287522a^22 + 40256/1804714587a^21 - 431963/10828287522a^20 + 468544/5414143761a^19 - 674903/3609429174a^18 + 1404757/10828287522a^17 - 1199045/1804714587a^16 + 21482105/10828287522a^15 - 28125599/10828287522a^14 + 6503944/1804714587a^13 - 81642577/10828287522a^12 + 142134637/10828287522a^11 - 19657337/1804714587a^10 + 40573889/1203143058a^9 - 10937405/133682562a^8 + 78160/825201a^7 - 5644045/44560854a^6 + 3457585/14853618a^5 - 288760/825201a^4 + 169573/1650402a^3 - 107527/183378a^2 + 97705/61126a - 27579/61126, 3787/3609429174a^23 - 3938/601571529a^22 - 31622/1804714587a^21 - 13679/3609429174a^20 + 17243/1203143058a^19 - 66637/3609429174a^18 + 336367/3609429174a^17 + 687889/3609429174a^16 + 471515/3609429174a^15 + 2196229/1203143058a^14 + 1450957/3609429174a^13 - 3115507/3609429174a^12 + 1365907/1203143058a^11 - 1996871/3609429174a^10 - 13613569/1203143058a^9 - 248549/44560854a^8 - 11873117/133682562a^7 - 48655/4951206a^6 - 544825/14853618a^5 - 11249/550134a^4 - 54803/1650402a^3 + 10393/91689a^2 + 22225/183378*a + 165747/61126 (assuming GRH)	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:	$ 55738390.68907589 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 55738390.68907589 \cdot 9}{42\cdot\sqrt{133084684332123489494188901166116961}}\cr\approx \mathstrut & 0.123948714022043 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 3*x^22 - 8*x^21 + 8*x^20 - 24*x^19 + 37*x^18 - 120*x^17 + 194*x^16 - 329*x^15 + 744*x^14 - 904*x^13 + 1633*x^12 - 2712*x^11 + 6696*x^10 - 8883*x^9 + 15714*x^8 - 29160*x^7 + 26973*x^6 - 52488*x^5 + 52488*x^4 - 157464*x^3 + 177147*x^2 - 177147*x + 531441)

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^24 - x^23 + 3*x^22 - 8*x^21 + 8*x^20 - 24*x^19 + 37*x^18 - 120*x^17 + 194*x^16 - 329*x^15 + 744*x^14 - 904*x^13 + 1633*x^12 - 2712*x^11 + 6696*x^10 - 8883*x^9 + 15714*x^8 - 29160*x^7 + 26973*x^6 - 52488*x^5 + 52488*x^4 - 157464*x^3 + 177147*x^2 - 177147*x + 531441, 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^24 - x^23 + 3*x^22 - 8*x^21 + 8*x^20 - 24*x^19 + 37*x^18 - 120*x^17 + 194*x^16 - 329*x^15 + 744*x^14 - 904*x^13 + 1633*x^12 - 2712*x^11 + 6696*x^10 - 8883*x^9 + 15714*x^8 - 29160*x^7 + 26973*x^6 - 52488*x^5 + 52488*x^4 - 157464*x^3 + 177147*x^2 - 177147*x + 531441);

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^24 - x^23 + 3*x^22 - 8*x^21 + 8*x^20 - 24*x^19 + 37*x^18 - 120*x^17 + 194*x^16 - 329*x^15 + 744*x^14 - 904*x^13 + 1633*x^12 - 2712*x^11 + 6696*x^10 - 8883*x^9 + 15714*x^8 - 29160*x^7 + 26973*x^6 - 52488*x^5 + 52488*x^4 - 157464*x^3 + 177147*x^2 - 177147*x + 531441);

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

$C_2^2\times C_6$ (as 24T3):

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)

An abelian group of order 24

The 24 conjugacy class representatives for $C_2^2\times C_6$

Character table for $C_2^2\times C_6$

Intermediate fields

$\Q(\sqrt{77}) $, $\Q(\sqrt{-231}) $, $\Q(\sqrt{-3}) $, $\Q(\sqrt{33}) $, $\Q(\sqrt{21}) $, $\Q(\sqrt{-7}) $, $\Q(\sqrt{-11}) $, $\Q(\zeta_{7})^+$, $\Q(\sqrt{-3}, \sqrt{77})$, $\Q(\sqrt{21}, \sqrt{33})$, $\Q(\sqrt{-7}, \sqrt{-11})$, $\Q(\sqrt{-7}, \sqrt{33})$, $\Q(\sqrt{-11}, \sqrt{21})$, $\Q(\sqrt{-3}, \sqrt{-11})$, $\Q(\sqrt{-3}, \sqrt{-7})$, 6.6.22370117.1, 6.0.603993159.1, 6.0.64827.1, 6.6.86284737.1, $\Q(\zeta_{21})^+$, $\Q(\zeta_{7})$, 6.0.3195731.1, 8.0.2847396321.1, 12.0.364807736118799281.1, 12.12.364807736118799281.1, 12.0.500422134593689.1, 12.0.364807736118799281.3, 12.0.364807736118799281.2, 12.0.7445055839159169.1, $\Q(\zeta_{21})$

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]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.6.0.1}{6} }^{4}$	R	${\href{/padicField/5.6.0.1}{6} }^{4}$	R	R	${\href{/padicField/13.2.0.1}{2} }^{12}$	${\href{/padicField/17.6.0.1}{6} }^{4}$	${\href{/padicField/19.6.0.1}{6} }^{4}$	${\href{/padicField/23.6.0.1}{6} }^{4}$	${\href{/padicField/29.2.0.1}{2} }^{12}$	${\href{/padicField/31.6.0.1}{6} }^{4}$	${\href{/padicField/37.3.0.1}{3} }^{8}$	${\href{/padicField/41.2.0.1}{2} }^{12}$	${\href{/padicField/43.2.0.1}{2} }^{12}$	${\href{/padicField/47.6.0.1}{6} }^{4}$	${\href{/padicField/53.6.0.1}{6} }^{4}$	${\href{/padicField/59.6.0.1}{6} }^{4}$

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	3.12.6.2	$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$3$ 3	3.12.6.2	$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$7$ 7	7.12.10.1	$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$	$6$	$2$	$10$	$C_6\times C_2$	$[\ ]_{6}^{2}$
$7$ 7	7.12.10.1	$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$	$6$	$2$	$10$	$C_6\times C_2$	$[\ ]_{6}^{2}$
$11$ 11	11.12.6.1	$x^{12} + 72 x^{10} + 8 x^{9} + 2034 x^{8} + 38 x^{7} + 27996 x^{6} - 6312 x^{5} + 196025 x^{4} - 84710 x^{3} + 695581 x^{2} - 235284 x + 1080083$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$11$ 11	11.12.6.1	$x^{12} + 72 x^{10} + 8 x^{9} + 2034 x^{8} + 38 x^{7} + 27996 x^{6} - 6312 x^{5} + 196025 x^{4} - 84710 x^{3} + 695581 x^{2} - 235284 x + 1080083$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$

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