Properties

Label 36.0.177...336.1
Degree $36$
Signature $(0, 18)$
Discriminant $1.777\times 10^{87}$
Root discriminant \(265.22\)
Ramified primes $2,3,7,47,67$
Class number not computed
Class group not computed
Galois group $C_2^2\times S_4^2$ (as 36T3161)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 73*x^34 - 66*x^33 + 3105*x^32 - 2649*x^31 + 88762*x^30 - 71975*x^29 + 1899311*x^28 - 1478352*x^27 + 31232683*x^26 - 23276523*x^25 + 406458419*x^24 - 286872742*x^23 + 4190952558*x^22 - 2720369448*x^21 + 34479543144*x^20 - 19495741960*x^19 + 223256681352*x^18 - 98727072288*x^17 + 1132339328928*x^16 - 316069561248*x^15 + 4386525309664*x^14 - 299570401728*x^13 + 13075210905536*x^12 + 1574606120192*x^11 + 29950092829952*x^10 + 8845110654976*x^9 + 51321732841472*x^8 + 19494359330816*x^7 + 62930183213056*x^6 + 25083388919808*x^5 + 46988599885824*x^4 + 9038185693184*x^3 + 14589905272832*x^2 - 17686593536*x + 3305531244544)
 
Copy content gp:K = bnfinit(y^36 - y^35 + 73*y^34 - 66*y^33 + 3105*y^32 - 2649*y^31 + 88762*y^30 - 71975*y^29 + 1899311*y^28 - 1478352*y^27 + 31232683*y^26 - 23276523*y^25 + 406458419*y^24 - 286872742*y^23 + 4190952558*y^22 - 2720369448*y^21 + 34479543144*y^20 - 19495741960*y^19 + 223256681352*y^18 - 98727072288*y^17 + 1132339328928*y^16 - 316069561248*y^15 + 4386525309664*y^14 - 299570401728*y^13 + 13075210905536*y^12 + 1574606120192*y^11 + 29950092829952*y^10 + 8845110654976*y^9 + 51321732841472*y^8 + 19494359330816*y^7 + 62930183213056*y^6 + 25083388919808*y^5 + 46988599885824*y^4 + 9038185693184*y^3 + 14589905272832*y^2 - 17686593536*y + 3305531244544, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - x^35 + 73*x^34 - 66*x^33 + 3105*x^32 - 2649*x^31 + 88762*x^30 - 71975*x^29 + 1899311*x^28 - 1478352*x^27 + 31232683*x^26 - 23276523*x^25 + 406458419*x^24 - 286872742*x^23 + 4190952558*x^22 - 2720369448*x^21 + 34479543144*x^20 - 19495741960*x^19 + 223256681352*x^18 - 98727072288*x^17 + 1132339328928*x^16 - 316069561248*x^15 + 4386525309664*x^14 - 299570401728*x^13 + 13075210905536*x^12 + 1574606120192*x^11 + 29950092829952*x^10 + 8845110654976*x^9 + 51321732841472*x^8 + 19494359330816*x^7 + 62930183213056*x^6 + 25083388919808*x^5 + 46988599885824*x^4 + 9038185693184*x^3 + 14589905272832*x^2 - 17686593536*x + 3305531244544);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^36 - x^35 + 73*x^34 - 66*x^33 + 3105*x^32 - 2649*x^31 + 88762*x^30 - 71975*x^29 + 1899311*x^28 - 1478352*x^27 + 31232683*x^26 - 23276523*x^25 + 406458419*x^24 - 286872742*x^23 + 4190952558*x^22 - 2720369448*x^21 + 34479543144*x^20 - 19495741960*x^19 + 223256681352*x^18 - 98727072288*x^17 + 1132339328928*x^16 - 316069561248*x^15 + 4386525309664*x^14 - 299570401728*x^13 + 13075210905536*x^12 + 1574606120192*x^11 + 29950092829952*x^10 + 8845110654976*x^9 + 51321732841472*x^8 + 19494359330816*x^7 + 62930183213056*x^6 + 25083388919808*x^5 + 46988599885824*x^4 + 9038185693184*x^3 + 14589905272832*x^2 - 17686593536*x + 3305531244544)
 

\( x^{36} - x^{35} + 73 x^{34} - 66 x^{33} + 3105 x^{32} - 2649 x^{31} + 88762 x^{30} + \cdots + 3305531244544 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $36$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(0, 18)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(177\!\cdots\!336\) \(\medspace = 2^{38}\cdot 3^{18}\cdot 7^{18}\cdot 47^{18}\cdot 67^{12}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(265.22\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{11/6}3^{1/2}7^{3/4}47^{3/4}67^{1/2}\approx 3902.862737609678$
Ramified primes:   \(2\), \(3\), \(7\), \(47\), \(67\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{13}+\frac{1}{4}a^{10}+\frac{1}{4}a^{7}-\frac{1}{4}a^{4}$, $\frac{1}{8}a^{14}-\frac{1}{4}a^{12}+\frac{1}{8}a^{11}-\frac{1}{2}a^{10}-\frac{1}{4}a^{9}+\frac{1}{8}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{8}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{15}+\frac{1}{8}a^{12}+\frac{1}{8}a^{9}-\frac{1}{8}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{16}-\frac{1}{8}a^{13}-\frac{1}{8}a^{10}-\frac{3}{8}a^{7}-\frac{1}{4}a^{4}$, $\frac{1}{8}a^{17}-\frac{1}{4}a^{12}-\frac{1}{2}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{3}{8}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{18}-\frac{1}{16}a^{16}-\frac{1}{16}a^{15}-\frac{1}{16}a^{13}-\frac{1}{16}a^{12}+\frac{7}{16}a^{10}-\frac{3}{16}a^{9}+\frac{1}{16}a^{7}+\frac{3}{8}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{19}-\frac{1}{16}a^{17}-\frac{1}{16}a^{16}-\frac{1}{16}a^{14}-\frac{1}{16}a^{13}-\frac{1}{16}a^{11}-\frac{3}{16}a^{10}-\frac{7}{16}a^{8}+\frac{3}{8}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{16}a^{20}-\frac{1}{16}a^{17}-\frac{1}{16}a^{16}-\frac{1}{16}a^{14}-\frac{1}{16}a^{13}-\frac{3}{16}a^{11}+\frac{7}{16}a^{10}-\frac{1}{2}a^{9}+\frac{3}{8}a^{8}-\frac{7}{16}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{21}-\frac{1}{32}a^{20}-\frac{1}{32}a^{19}-\frac{1}{32}a^{17}-\frac{1}{32}a^{16}-\frac{1}{32}a^{14}-\frac{1}{32}a^{13}+\frac{3}{16}a^{12}+\frac{1}{32}a^{11}-\frac{7}{32}a^{10}+\frac{5}{32}a^{9}+\frac{1}{8}a^{8}+\frac{3}{8}a^{7}-\frac{1}{8}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}$, $\frac{1}{64}a^{22}-\frac{1}{64}a^{21}-\frac{1}{64}a^{20}-\frac{1}{64}a^{18}+\frac{3}{64}a^{17}+\frac{3}{64}a^{15}+\frac{3}{64}a^{14}-\frac{1}{32}a^{13}-\frac{11}{64}a^{12}+\frac{13}{64}a^{11}-\frac{3}{64}a^{10}+\frac{3}{8}a^{9}-\frac{1}{8}a^{8}-\frac{3}{16}a^{7}+\frac{5}{16}a^{6}+\frac{3}{8}a^{5}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{23}+\frac{1}{64}a^{20}+\frac{1}{64}a^{19}-\frac{1}{32}a^{18}+\frac{1}{64}a^{17}-\frac{3}{64}a^{16}+\frac{1}{32}a^{15}-\frac{1}{64}a^{14}-\frac{3}{64}a^{13}+\frac{5}{32}a^{12}+\frac{1}{16}a^{11}+\frac{11}{64}a^{10}-\frac{1}{32}a^{9}-\frac{3}{8}a^{8}-\frac{3}{8}a^{7}+\frac{5}{16}a^{6}-\frac{1}{8}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{384}a^{24}+\frac{1}{384}a^{23}+\frac{1}{384}a^{22}-\frac{1}{192}a^{21}+\frac{7}{384}a^{20}-\frac{11}{384}a^{19}+\frac{5}{192}a^{18}-\frac{13}{384}a^{17}+\frac{7}{128}a^{16}+\frac{1}{48}a^{15}-\frac{5}{128}a^{14}-\frac{37}{384}a^{13}+\frac{91}{384}a^{12}+\frac{17}{192}a^{11}-\frac{1}{4}a^{9}+\frac{41}{96}a^{8}-\frac{1}{16}a^{7}-\frac{7}{24}a^{5}-\frac{5}{24}a^{4}+\frac{1}{6}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{384}a^{25}-\frac{1}{128}a^{22}-\frac{1}{128}a^{21}-\frac{1}{64}a^{20}+\frac{3}{128}a^{19}+\frac{1}{384}a^{18}+\frac{11}{192}a^{17}-\frac{1}{384}a^{16}+\frac{1}{384}a^{15}+\frac{7}{192}a^{14}+\frac{11}{96}a^{13}-\frac{3}{128}a^{12}-\frac{11}{192}a^{11}-\frac{5}{32}a^{10}-\frac{7}{24}a^{9}+\frac{7}{96}a^{8}-\frac{3}{8}a^{7}-\frac{1}{6}a^{6}+\frac{11}{24}a^{5}-\frac{3}{8}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{768}a^{26}-\frac{1}{768}a^{25}-\frac{1}{768}a^{24}-\frac{1}{192}a^{23}-\frac{1}{768}a^{22}+\frac{11}{768}a^{21}-\frac{1}{192}a^{20}-\frac{3}{256}a^{19}-\frac{13}{768}a^{18}+\frac{13}{384}a^{17}+\frac{41}{768}a^{16}-\frac{19}{768}a^{15}+\frac{11}{256}a^{14}-\frac{13}{192}a^{13}+\frac{5}{96}a^{12}+\frac{11}{64}a^{11}+\frac{7}{24}a^{10}+\frac{1}{24}a^{9}-\frac{1}{4}a^{8}+\frac{17}{48}a^{7}-\frac{1}{4}a^{6}-\frac{1}{3}a^{5}-\frac{1}{12}a^{4}-\frac{1}{4}a^{2}+\frac{1}{3}a$, $\frac{1}{1536}a^{27}-\frac{1}{1536}a^{26}+\frac{1}{1536}a^{25}-\frac{1}{768}a^{24}+\frac{1}{1536}a^{23}+\frac{7}{1536}a^{22}+\frac{5}{768}a^{21}-\frac{31}{1536}a^{20}+\frac{7}{1536}a^{19}-\frac{13}{1536}a^{17}+\frac{31}{512}a^{16}-\frac{31}{512}a^{15}-\frac{5}{256}a^{14}+\frac{21}{256}a^{13}-\frac{35}{192}a^{12}-\frac{1}{24}a^{11}-\frac{1}{6}a^{10}-\frac{43}{192}a^{9}+\frac{1}{12}a^{8}+\frac{3}{16}a^{7}-\frac{1}{2}a^{6}-\frac{13}{48}a^{5}+\frac{11}{24}a^{4}+\frac{5}{24}a^{3}+\frac{1}{12}a^{2}+\frac{1}{3}a$, $\frac{1}{3072}a^{28}-\frac{1}{3072}a^{27}+\frac{1}{3072}a^{26}-\frac{1}{1536}a^{25}+\frac{1}{3072}a^{24}-\frac{17}{3072}a^{23}-\frac{7}{1536}a^{22}+\frac{41}{3072}a^{21}+\frac{55}{3072}a^{20}-\frac{3}{128}a^{19}-\frac{37}{3072}a^{18}-\frac{49}{1024}a^{17}-\frac{23}{1024}a^{16}-\frac{9}{512}a^{15}-\frac{11}{512}a^{14}-\frac{13}{192}a^{13}+\frac{43}{384}a^{12}+\frac{79}{384}a^{11}+\frac{83}{384}a^{10}-\frac{5}{24}a^{9}-\frac{13}{32}a^{8}+\frac{15}{32}a^{7}-\frac{43}{96}a^{6}-\frac{1}{48}a^{5}+\frac{23}{48}a^{4}+\frac{5}{12}a^{3}+\frac{1}{6}a^{2}-\frac{1}{2}a$, $\frac{1}{6144}a^{29}-\frac{1}{6144}a^{28}+\frac{1}{6144}a^{27}-\frac{1}{3072}a^{26}+\frac{1}{6144}a^{25}-\frac{1}{6144}a^{24}-\frac{23}{3072}a^{23}+\frac{3}{2048}a^{22}-\frac{25}{6144}a^{21}-\frac{7}{768}a^{20}-\frac{55}{2048}a^{19}-\frac{35}{6144}a^{18}+\frac{203}{6144}a^{17}+\frac{23}{1024}a^{16}+\frac{7}{3072}a^{15}+\frac{1}{48}a^{14}-\frac{37}{768}a^{13}+\frac{9}{256}a^{12}+\frac{157}{768}a^{11}+\frac{73}{192}a^{10}+\frac{25}{64}a^{9}-\frac{77}{192}a^{8}+\frac{23}{192}a^{7}+\frac{5}{96}a^{6}+\frac{19}{96}a^{5}+\frac{3}{8}a^{4}+\frac{1}{8}a^{3}-\frac{1}{12}a^{2}-\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{24576}a^{30}+\frac{1}{24576}a^{29}+\frac{1}{8192}a^{28}+\frac{1}{6144}a^{27}-\frac{7}{24576}a^{26}+\frac{17}{24576}a^{25}-\frac{1}{2048}a^{24}-\frac{95}{24576}a^{23}+\frac{185}{24576}a^{22}-\frac{1}{4096}a^{21}+\frac{703}{24576}a^{20}-\frac{135}{8192}a^{19}-\frac{479}{24576}a^{18}-\frac{99}{2048}a^{17}-\frac{181}{12288}a^{16}+\frac{43}{6144}a^{15}-\frac{65}{1536}a^{14}+\frac{99}{1024}a^{13}-\frac{449}{3072}a^{12}-\frac{259}{1536}a^{11}-\frac{25}{128}a^{10}-\frac{215}{768}a^{9}-\frac{355}{768}a^{8}+\frac{47}{192}a^{7}+\frac{35}{384}a^{6}-\frac{17}{192}a^{5}-\frac{5}{16}a^{4}+\frac{17}{48}a^{3}+\frac{1}{12}a^{2}-\frac{5}{12}a+\frac{1}{6}$, $\frac{1}{3293184}a^{31}+\frac{19}{1097728}a^{30}-\frac{149}{3293184}a^{29}+\frac{37}{274432}a^{28}+\frac{457}{3293184}a^{27}-\frac{349}{1097728}a^{26}-\frac{195}{274432}a^{25}-\frac{4079}{3293184}a^{24}-\frac{1775}{3293184}a^{23}+\frac{2139}{548864}a^{22}-\frac{46081}{3293184}a^{21}+\frac{17651}{3293184}a^{20}-\frac{45287}{3293184}a^{19}+\frac{5617}{823296}a^{18}+\frac{18115}{1646592}a^{17}-\frac{1369}{823296}a^{16}-\frac{1383}{68608}a^{15}-\frac{23863}{411648}a^{14}-\frac{13081}{411648}a^{13}+\frac{6403}{68608}a^{12}+\frac{5947}{51456}a^{11}+\frac{3403}{34304}a^{10}+\frac{16693}{102912}a^{9}-\frac{2709}{8576}a^{8}+\frac{145}{768}a^{7}+\frac{1705}{8576}a^{6}+\frac{2221}{6432}a^{5}-\frac{25}{2144}a^{4}+\frac{80}{201}a^{3}+\frac{703}{1608}a^{2}+\frac{43}{268}a$, $\frac{1}{401768448}a^{32}+\frac{1}{401768448}a^{31}+\frac{4967}{401768448}a^{30}-\frac{1079}{50221056}a^{29}-\frac{1627}{401768448}a^{28}-\frac{29855}{401768448}a^{27}-\frac{3181}{50221056}a^{26}-\frac{4937}{133922816}a^{25}+\frac{84483}{133922816}a^{24}+\frac{182677}{66961408}a^{23}-\frac{2889757}{401768448}a^{22}+\frac{1874993}{133922816}a^{21}+\frac{132767}{5996544}a^{20}-\frac{242453}{16740352}a^{19}-\frac{364865}{66961408}a^{18}+\frac{2545271}{100442112}a^{17}+\frac{2730469}{50221056}a^{16}+\frac{627421}{16740352}a^{15}-\frac{898203}{16740352}a^{14}-\frac{675401}{25110528}a^{13}-\frac{2863387}{12555264}a^{12}+\frac{83105}{4185088}a^{11}+\frac{1545037}{12555264}a^{10}-\frac{191459}{784704}a^{9}+\frac{1894789}{6277632}a^{8}+\frac{681011}{3138816}a^{7}+\frac{669041}{1569408}a^{6}+\frac{16145}{49044}a^{5}+\frac{15451}{98088}a^{4}+\frac{4343}{24522}a^{3}-\frac{39923}{98088}a^{2}+\frac{5701}{49044}a-\frac{37}{366}$, $\frac{1}{553636921344}a^{33}-\frac{7}{8263237632}a^{32}-\frac{20949}{184545640448}a^{31}-\frac{3924613}{276818460672}a^{30}+\frac{7061567}{184545640448}a^{29}+\frac{10690523}{553636921344}a^{28}-\frac{23676813}{92272820224}a^{27}+\frac{55091295}{184545640448}a^{26}+\frac{27828209}{184545640448}a^{25}-\frac{88051331}{69204615168}a^{24}-\frac{2328450089}{553636921344}a^{23}+\frac{1277832099}{184545640448}a^{22}+\frac{1575847643}{553636921344}a^{21}+\frac{3058373761}{276818460672}a^{20}+\frac{4752087985}{276818460672}a^{19}-\frac{854410709}{69204615168}a^{18}-\frac{266029105}{5767051264}a^{17}+\frac{3969984611}{69204615168}a^{16}-\frac{11448011}{5323431936}a^{15}+\frac{138467383}{5767051264}a^{14}-\frac{291153489}{2883525632}a^{13}-\frac{82991521}{17301153792}a^{12}-\frac{4119146705}{17301153792}a^{11}-\frac{1039321727}{8650576896}a^{10}+\frac{3458213545}{8650576896}a^{9}+\frac{117349331}{720881408}a^{8}+\frac{25402743}{180220352}a^{7}+\frac{43623283}{180220352}a^{6}-\frac{11660903}{180220352}a^{5}+\frac{473851}{67582632}a^{4}+\frac{14318063}{67582632}a^{3}+\frac{10996079}{33791316}a^{2}-\frac{3935527}{11263772}a-\frac{166}{2379}$, $\frac{1}{14\cdots 28}a^{34}+\frac{734773417}{14\cdots 28}a^{33}-\frac{207013477227}{49\cdots 76}a^{32}+\frac{47993063947}{46\cdots 04}a^{31}+\frac{61\cdots 51}{49\cdots 76}a^{30}+\frac{22\cdots 05}{14\cdots 28}a^{29}-\frac{71625670643927}{44\cdots 76}a^{28}+\frac{47\cdots 49}{14\cdots 28}a^{27}+\frac{32\cdots 95}{49\cdots 76}a^{26}-\frac{96\cdots 65}{74\cdots 64}a^{25}+\frac{81\cdots 71}{14\cdots 28}a^{24}-\frac{91\cdots 77}{14\cdots 28}a^{23}-\frac{24\cdots 61}{49\cdots 76}a^{22}-\frac{41\cdots 91}{31\cdots 36}a^{21}-\frac{14\cdots 39}{74\cdots 64}a^{20}-\frac{76\cdots 01}{37\cdots 32}a^{19}-\frac{45\cdots 45}{18\cdots 16}a^{18}+\frac{96\cdots 49}{27\cdots 48}a^{17}+\frac{18\cdots 73}{62\cdots 72}a^{16}+\frac{49\cdots 83}{93\cdots 08}a^{15}-\frac{59\cdots 65}{15\cdots 68}a^{14}+\frac{15\cdots 45}{15\cdots 68}a^{13}-\frac{11\cdots 57}{15\cdots 68}a^{12}-\frac{37\cdots 77}{22\cdots 88}a^{11}+\frac{17\cdots 99}{77\cdots 84}a^{10}+\frac{18\cdots 63}{11\cdots 76}a^{9}+\frac{47\cdots 63}{11\cdots 96}a^{8}+\frac{14\cdots 09}{48\cdots 24}a^{7}-\frac{54\cdots 57}{14\cdots 72}a^{6}-\frac{14\cdots 33}{73\cdots 36}a^{5}-\frac{259632822322411}{54\cdots 04}a^{4}-\frac{22\cdots 49}{45\cdots 96}a^{3}-\frac{45\cdots 83}{91\cdots 92}a^{2}+\frac{332587374679855}{681976947755688}a-\frac{23678152729}{96026041644}$, $\frac{1}{27\cdots 04}a^{35}-\frac{56\cdots 79}{27\cdots 04}a^{34}+\frac{19\cdots 47}{52\cdots 68}a^{33}-\frac{20\cdots 75}{23\cdots 92}a^{32}-\frac{12\cdots 61}{92\cdots 68}a^{31}-\frac{13\cdots 23}{27\cdots 04}a^{30}+\frac{19\cdots 09}{69\cdots 76}a^{29}-\frac{46\cdots 77}{71\cdots 36}a^{28}+\frac{33\cdots 73}{27\cdots 04}a^{27}+\frac{86\cdots 57}{13\cdots 52}a^{26}+\frac{73\cdots 65}{92\cdots 68}a^{25}-\frac{15\cdots 25}{27\cdots 04}a^{24}+\frac{48\cdots 03}{92\cdots 68}a^{23}-\frac{11\cdots 11}{53\cdots 52}a^{22}+\frac{62\cdots 97}{13\cdots 52}a^{21}+\frac{46\cdots 95}{23\cdots 92}a^{20}-\frac{10\cdots 17}{34\cdots 88}a^{19}+\frac{33\cdots 95}{34\cdots 88}a^{18}+\frac{14\cdots 67}{34\cdots 88}a^{17}-\frac{31\cdots 91}{17\cdots 44}a^{16}+\frac{47\cdots 69}{86\cdots 72}a^{15}-\frac{37\cdots 05}{86\cdots 72}a^{14}-\frac{38\cdots 11}{86\cdots 72}a^{13}-\frac{11\cdots 15}{21\cdots 68}a^{12}-\frac{25\cdots 05}{14\cdots 12}a^{11}+\frac{62\cdots 01}{21\cdots 68}a^{10}-\frac{99\cdots 75}{36\cdots 28}a^{9}+\frac{63\cdots 51}{13\cdots 48}a^{8}-\frac{31\cdots 41}{90\cdots 32}a^{7}+\frac{90\cdots 85}{45\cdots 16}a^{6}+\frac{56\cdots 01}{22\cdots 08}a^{5}-\frac{32\cdots 89}{77\cdots 18}a^{4}+\frac{17\cdots 31}{56\cdots 52}a^{3}+\frac{10\cdots 13}{84\cdots 28}a^{2}+\frac{12\cdots 59}{46\cdots 16}a-\frac{17\cdots 83}{14\cdots 66}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  not computed
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  not computed
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 
Relative class number:   data not computed

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $17$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( \frac{52061581005684957565978666380877751583983723403688660619467189165766762476500014275176339002359232651611157081227327930633519}{3066242737385496668417018132567615403573094713609888590623187104344377580295669928127038070567250882044177234615769911520077177443258302464} a^{35} - \frac{450223518853652477103549079402229870199979244655718386216589776388586633724054144929306600717082543668163262828985617162607627}{3066242737385496668417018132567615403573094713609888590623187104344377580295669928127038070567250882044177234615769911520077177443258302464} a^{34} + \frac{4082230602231193351523420531357364672755783131475424790183098937994892644966276241993117869321736551303659384020428853687787745}{3066242737385496668417018132567615403573094713609888590623187104344377580295669928127038070567250882044177234615769911520077177443258302464} a^{33} - \frac{16114607642934995169375784352740188036514134468506857006064895681656309200460238735147563086830249334304340543586516624836141161}{1533121368692748334208509066283807701786547356804944295311593552172188790147834964063519035283625441022088617307884955760038588721629151232} a^{32} + \frac{59827202241732162220481563271522996455643091838470541640720392851106634635415438810822889834651534537033846203615523617041053533}{1022080912461832222805672710855871801191031571203296196874395701448125860098556642709012690189083627348059078205256637173359059147752767488} a^{31} - \frac{1355459392328897655102312360417210382838839705507461418284208960220610540331748896528643747140608051526409003494473062928304175881}{3066242737385496668417018132567615403573094713609888590623187104344377580295669928127038070567250882044177234615769911520077177443258302464} a^{30} + \frac{1329719316442914479109092582174613222350121734394033647833951952444565911648278487854669313708928605622573415360837421544147746881}{766560684346374167104254533141903850893273678402472147655796776086094395073917482031759517641812720511044308653942477880019294360814575616} a^{29} - \frac{38318461868011042098227452151631350564226239781450045441615291110629763994671906716751800336062914502566628888448586441906073091873}{3066242737385496668417018132567615403573094713609888590623187104344377580295669928127038070567250882044177234615769911520077177443258302464} a^{28} + \frac{117415588168611180804988128227600579032443799435312603062868772906896013461863363666350874713946362424332760554717010929294144015795}{3066242737385496668417018132567615403573094713609888590623187104344377580295669928127038070567250882044177234615769911520077177443258302464} a^{27} - \frac{135315561205754279564752833695678207668396547223656728622385602027257046927442940293025071119933085612898091688293597840582965846223}{511040456230916111402836355427935900595515785601648098437197850724062930049278321354506345094541813674029539102628318586679529573876383744} a^{26} + \frac{1999880277588963096094653012211929163935070703303539884484695809431887164426697368901366011990292219584834204709983154946752205963925}{3066242737385496668417018132567615403573094713609888590623187104344377580295669928127038070567250882044177234615769911520077177443258302464} a^{25} - \frac{4401461077431953922254927535341743969803496951589088728355920980795516188495625298009386082445790787598129691588931341312645671044749}{1022080912461832222805672710855871801191031571203296196874395701448125860098556642709012690189083627348059078205256637173359059147752767488} a^{24} + \frac{8977607798063142258240301823121725193552505798413025969541218046008076552889143109432309449284549869366288029508698478670888305166073}{1022080912461832222805672710855871801191031571203296196874395701448125860098556642709012690189083627348059078205256637173359059147752767488} a^{23} - \frac{84856257843082887745933945593266285873263684714633532336015179978147457962803875914965713766783778700609317048309909694089271154642887}{1533121368692748334208509066283807701786547356804944295311593552172188790147834964063519035283625441022088617307884955760038588721629151232} a^{22} + \frac{143727524551476906962545022623848795225458596925415947769274313492969388626786527680199945527698852648951372469721672291939149621518069}{1533121368692748334208509066283807701786547356804944295311593552172188790147834964063519035283625441022088617307884955760038588721629151232} a^{21} - \frac{860885928148225910034467039385538237784324194796122869367553143520685597056185721054214217161662334735565237035839999975813387017438169}{1533121368692748334208509066283807701786547356804944295311593552172188790147834964063519035283625441022088617307884955760038588721629151232} a^{20} + \frac{302766803976594059093640102975817985836062211332828766726678731737622366483605633813043373318482324447416836618125424442538137681521861}{383280342173187083552127266570951925446636839201236073827898388043047197536958741015879758820906360255522154326971238940009647180407287808} a^{19} - \frac{3468906250893542847436738916303702698419024938330022828104381175395100289326820929332712647150429543374851008630367797033680157107666031}{766560684346374167104254533141903850893273678402472147655796776086094395073917482031759517641812720511044308653942477880019294360814575616} a^{18} + \frac{654925776918377984682414770140702020675151779178908485198152061504112140735969553233995528005120615240151499554814368679301770289115621}{127760114057729027850709088856983975148878946400412024609299462681015732512319580338626586273635453418507384775657079646669882393469095936} a^{17} - \frac{5460857862159293040522376749997194318617834565078236815140681219864322175620899263294054533057681374067860483818957582457609643652418037}{191640171086593541776063633285475962723318419600618036913949194021523598768479370507939879410453180127761077163485619470004823590203643904} a^{16} + \frac{2355870458264186753180293169633342535286825035232764874370342567794868065153638405200174610241171864708296153158311904152718197603579297}{95820085543296770888031816642737981361659209800309018456974597010761799384239685253969939705226590063880538581742809735002411795101821952} a^{15} - \frac{26774185992439373076261814871488089486537028494887798526147010327516964818164912343000367628441956598155279365033134148830835408796123315}{191640171086593541776063633285475962723318419600618036913949194021523598768479370507939879410453180127761077163485619470004823590203643904} a^{14} + \frac{7663208946184044649780665705861604372037298366950536814762595188909188210525597150354934582522734275622709189472565645270002839021978113}{95820085543296770888031816642737981361659209800309018456974597010761799384239685253969939705226590063880538581742809735002411795101821952} a^{13} - \frac{6211189780609887298269705504311913865978486242160270987453074136228846112927593065572040727208301852298155074648431898975498360755457645}{11977510692912096361003977080342247670207401225038627307121824626345224923029960656746242463153323757985067322717851216875301474387727744} a^{12} + \frac{6605659429908768842899194553364837467700797771822309465302624432149469160770131627709890751349201039778994978813839804707810484171812063}{47910042771648385444015908321368990680829604900154509228487298505380899692119842626984969852613295031940269290871404867501205897550910976} a^{11} - \frac{71493252316528697714922547308121200996545826253582712576561565360851486138973574049792278067747209530956523576586784693440957557171086089}{47910042771648385444015908321368990680829604900154509228487298505380899692119842626984969852613295031940269290871404867501205897550910976} a^{10} + \frac{9434049725416518997043848787866827195576911390515917274432710078783637865947106753729007881220748946159833410175522429172381760529039}{3992503564304032120334659026780749223402467075012875769040608208781741641009986885582080821051107919328355774239283738958433824795909248} a^{9} - \frac{78828999639760383968529134884365304934616012259039791653251807076022528631525468537210483384091476429538057897427154524134905763420710265}{23955021385824192722007954160684495340414802450077254614243649252690449846059921313492484926306647515970134645435702433750602948775455488} a^{8} - \frac{294520601535383812000423260550107703093087031338924639483939224241335220999368746463602442482035512219688758864484839646998130188604271}{374297209153503011281374283760695239693981288282457103347557019573288278844686270523320076973541367437033353834932850527353171074616492} a^{7} - \frac{5490613516500862954601527749209397204021421100413605114085605641432850982563200103470242708023696009013578525131804101926539213740598643}{998125891076008030083664756695187305850616768753218942260152052195435410252496721395520205262776979832088943559820934739608456198977312} a^{6} - \frac{1642873250101824562563142238583375810428151641076456127165763262374097825958191489003466159553150118184183154413117673987999725194417}{854559838250006875071630784841769953639226685576386080702184976194722097818918425852328942861966592321993958527243950975692171403234} a^{5} - \frac{4822300386001707792427835393401192267088008405440549304017016820807993123917547224165297492087977024492699255282656217299648715719230871}{748594418307006022562748567521390479387962576564914206695114039146576557689372541046640153947082734874066707669865701054706342149232984} a^{4} - \frac{79287370107043323801032250089374445008528044134118685032046092342151144538339823107081161928776742713429329922808633924921207739191851}{31191434096125250940114523646724603307831774023538091945629751631107356570390522543610006414461780619752779486244404210612764256218041} a^{3} - \frac{302967778129286193942326731706064810935630047972041032842825302594078754465639262087183289762444900215815678439484927979017139817620407}{62382868192250501880229047293449206615663548047076183891259503262214713140781045087220012828923561239505558972488808421225528512436082} a^{2} - \frac{37717489806853537466215161095768543995521057137381976833934930067543074440820361125786897527465923141272695348564256180293632753006210}{93574302288375752820343570940173809923495322070614275836889254893322069711171567630830019243385341859258338458733212631838292768654123} a - \frac{13114204452681047782101089982341048963259607512999395898858471622985799134233277627739471620090943970022901367226488901462242205938}{26351535423366869281989177961186654441986854990316608233424177666381883895007481732140247604445322967969118124115238702291831249973} \)  (order $6$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:  not computed
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot R \cdot h}{6\cdot\sqrt{1777240558318257494226779210162588854257424158936371353076420013776262325086096338190336}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 73*x^34 - 66*x^33 + 3105*x^32 - 2649*x^31 + 88762*x^30 - 71975*x^29 + 1899311*x^28 - 1478352*x^27 + 31232683*x^26 - 23276523*x^25 + 406458419*x^24 - 286872742*x^23 + 4190952558*x^22 - 2720369448*x^21 + 34479543144*x^20 - 19495741960*x^19 + 223256681352*x^18 - 98727072288*x^17 + 1132339328928*x^16 - 316069561248*x^15 + 4386525309664*x^14 - 299570401728*x^13 + 13075210905536*x^12 + 1574606120192*x^11 + 29950092829952*x^10 + 8845110654976*x^9 + 51321732841472*x^8 + 19494359330816*x^7 + 62930183213056*x^6 + 25083388919808*x^5 + 46988599885824*x^4 + 9038185693184*x^3 + 14589905272832*x^2 - 17686593536*x + 3305531244544) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^36 - x^35 + 73*x^34 - 66*x^33 + 3105*x^32 - 2649*x^31 + 88762*x^30 - 71975*x^29 + 1899311*x^28 - 1478352*x^27 + 31232683*x^26 - 23276523*x^25 + 406458419*x^24 - 286872742*x^23 + 4190952558*x^22 - 2720369448*x^21 + 34479543144*x^20 - 19495741960*x^19 + 223256681352*x^18 - 98727072288*x^17 + 1132339328928*x^16 - 316069561248*x^15 + 4386525309664*x^14 - 299570401728*x^13 + 13075210905536*x^12 + 1574606120192*x^11 + 29950092829952*x^10 + 8845110654976*x^9 + 51321732841472*x^8 + 19494359330816*x^7 + 62930183213056*x^6 + 25083388919808*x^5 + 46988599885824*x^4 + 9038185693184*x^3 + 14589905272832*x^2 - 17686593536*x + 3305531244544, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - x^35 + 73*x^34 - 66*x^33 + 3105*x^32 - 2649*x^31 + 88762*x^30 - 71975*x^29 + 1899311*x^28 - 1478352*x^27 + 31232683*x^26 - 23276523*x^25 + 406458419*x^24 - 286872742*x^23 + 4190952558*x^22 - 2720369448*x^21 + 34479543144*x^20 - 19495741960*x^19 + 223256681352*x^18 - 98727072288*x^17 + 1132339328928*x^16 - 316069561248*x^15 + 4386525309664*x^14 - 299570401728*x^13 + 13075210905536*x^12 + 1574606120192*x^11 + 29950092829952*x^10 + 8845110654976*x^9 + 51321732841472*x^8 + 19494359330816*x^7 + 62930183213056*x^6 + 25083388919808*x^5 + 46988599885824*x^4 + 9038185693184*x^3 + 14589905272832*x^2 - 17686593536*x + 3305531244544); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^36 - x^35 + 73*x^34 - 66*x^33 + 3105*x^32 - 2649*x^31 + 88762*x^30 - 71975*x^29 + 1899311*x^28 - 1478352*x^27 + 31232683*x^26 - 23276523*x^25 + 406458419*x^24 - 286872742*x^23 + 4190952558*x^22 - 2720369448*x^21 + 34479543144*x^20 - 19495741960*x^19 + 223256681352*x^18 - 98727072288*x^17 + 1132339328928*x^16 - 316069561248*x^15 + 4386525309664*x^14 - 299570401728*x^13 + 13075210905536*x^12 + 1574606120192*x^11 + 29950092829952*x^10 + 8845110654976*x^9 + 51321732841472*x^8 + 19494359330816*x^7 + 62930183213056*x^6 + 25083388919808*x^5 + 46988599885824*x^4 + 9038185693184*x^3 + 14589905272832*x^2 - 17686593536*x + 3305531244544); 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

$C_2^2\times S_4^2$ (as 36T3161):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 2304
The 100 conjugacy class representatives for $C_2^2\times S_4^2$
Character table for $C_2^2\times S_4^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.564.1, 3.3.469.1, 6.0.5938947.1, 6.0.954288.1, 9.9.18507844420140096.1, 18.0.9248588237162993931281203206008832.1, 18.18.1561382630189567606481071728056448250806272.1, 18.0.42157331015118325374988936657524102771769344.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 36 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 R R ${\href{/padicField/5.6.0.1}{6} }^{6}$ R ${\href{/padicField/11.6.0.1}{6} }^{6}$ ${\href{/padicField/13.12.0.1}{12} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{10}$ ${\href{/padicField/19.4.0.1}{4} }^{6}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{6}$ ${\href{/padicField/31.12.0.1}{12} }^{2}{,}\,{\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{6}$ ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ R ${\href{/padicField/53.4.0.1}{4} }^{6}{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.2.0.1}{2} }^{18}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.6.22a1.9$x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 145 x^{6} + 138 x^{5} + 114 x^{4} + 78 x^{3} + 45 x^{2} + 18 x + 7$$6$$2$$22$$D_6$$$[3]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.3.2.3a1.2$x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
3.3.2.3a1.2$x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
3.6.2.6a1.2$x^{12} + 4 x^{10} + 6 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} + 9 x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
3.6.2.6a1.2$x^{12} + 4 x^{10} + 6 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} + 9 x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
\(7\) Copy content Toggle raw display 7.3.1.0a1.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$$[\ ]^{3}$$
7.3.1.0a1.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$$[\ ]^{3}$$
7.3.1.0a1.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$$[\ ]^{3}$$
7.3.1.0a1.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$$[\ ]^{3}$$
7.3.4.9a1.1$x^{12} + 24 x^{11} + 216 x^{10} + 880 x^{9} + 1584 x^{8} + 1728 x^{7} + 3552 x^{6} + 1152 x^{5} + 3456 x^{4} + 256 x^{3} + 1536 x^{2} + 7 x + 256$$4$$3$$9$$D_4 \times C_3$$$[\ ]_{4}^{6}$$
7.3.4.9a1.1$x^{12} + 24 x^{11} + 216 x^{10} + 880 x^{9} + 1584 x^{8} + 1728 x^{7} + 3552 x^{6} + 1152 x^{5} + 3456 x^{4} + 256 x^{3} + 1536 x^{2} + 7 x + 256$$4$$3$$9$$D_4 \times C_3$$$[\ ]_{4}^{6}$$
\(47\) Copy content Toggle raw display 47.2.1.0a1.1$x^{2} + 45 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
47.2.1.0a1.1$x^{2} + 45 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
47.4.1.0a1.1$x^{4} + 8 x^{2} + 40 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
47.4.1.0a1.1$x^{4} + 8 x^{2} + 40 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
47.2.4.6a1.2$x^{8} + 180 x^{7} + 12170 x^{6} + 367200 x^{5} + 4222275 x^{4} + 1836000 x^{3} + 304250 x^{2} + 22500 x + 672$$4$$2$$6$$D_4$$$[\ ]_{4}^{2}$$
47.2.4.6a1.3$x^{8} + 180 x^{7} + 12170 x^{6} + 367200 x^{5} + 4222275 x^{4} + 1836000 x^{3} + 304250 x^{2} + 22594 x + 2599$$4$$2$$6$$Q_8$$$[\ ]_{4}^{2}$$
47.2.4.6a1.3$x^{8} + 180 x^{7} + 12170 x^{6} + 367200 x^{5} + 4222275 x^{4} + 1836000 x^{3} + 304250 x^{2} + 22594 x + 2599$$4$$2$$6$$Q_8$$$[\ ]_{4}^{2}$$
\(67\) Copy content Toggle raw display $\Q_{67}$$x + 65$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{67}$$x + 65$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{67}$$x + 65$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{67}$$x + 65$$1$$1$$0$Trivial$$[\ ]$$
67.1.2.1a1.1$x^{2} + 67$$2$$1$$1$$C_2$$$[\ ]_{2}$$
67.2.1.0a1.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
67.1.2.1a1.1$x^{2} + 67$$2$$1$$1$$C_2$$$[\ ]_{2}$$
67.2.1.0a1.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
67.1.2.1a1.1$x^{2} + 67$$2$$1$$1$$C_2$$$[\ ]_{2}$$
67.1.2.1a1.1$x^{2} + 67$$2$$1$$1$$C_2$$$[\ ]_{2}$$
67.2.1.0a1.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
67.2.1.0a1.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
67.2.2.2a1.2$x^{4} + 126 x^{3} + 3973 x^{2} + 252 x + 71$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
67.2.2.2a1.2$x^{4} + 126 x^{3} + 3973 x^{2} + 252 x + 71$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
67.2.2.2a1.2$x^{4} + 126 x^{3} + 3973 x^{2} + 252 x + 71$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
67.2.2.2a1.2$x^{4} + 126 x^{3} + 3973 x^{2} + 252 x + 71$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)