Properties

Label 36.0.173...576.1
Degree $36$
Signature $(0, 18)$
Discriminant $1.730\times 10^{79}$
Root discriminant \(158.88\)
Ramified primes $2,3,7,67,199$
Class number not computed
Class group not computed
Galois group $A_4^2:C_2^3$ (as 36T1803)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^36 - 3*x^34 - 2*x^33 - 9*x^32 - 27*x^31 + 71*x^30 - 6*x^29 + 524*x^27 + 1812*x^26 - 1104*x^25 - 2280*x^24 - 15024*x^23 - 21696*x^22 - 3264*x^21 + 97152*x^20 + 16896*x^19 + 301568*x^18 + 67584*x^17 + 1554432*x^16 - 208896*x^15 - 5554176*x^14 - 15384576*x^13 - 9338880*x^12 - 18087936*x^11 + 118751232*x^10 + 137363456*x^9 - 25165824*x^7 + 1191182336*x^6 - 1811939328*x^5 - 2415919104*x^4 - 2147483648*x^3 - 12884901888*x^2 + 68719476736)
 
Copy content gp:K = bnfinit(y^36 - 3*y^34 - 2*y^33 - 9*y^32 - 27*y^31 + 71*y^30 - 6*y^29 + 524*y^27 + 1812*y^26 - 1104*y^25 - 2280*y^24 - 15024*y^23 - 21696*y^22 - 3264*y^21 + 97152*y^20 + 16896*y^19 + 301568*y^18 + 67584*y^17 + 1554432*y^16 - 208896*y^15 - 5554176*y^14 - 15384576*y^13 - 9338880*y^12 - 18087936*y^11 + 118751232*y^10 + 137363456*y^9 - 25165824*y^7 + 1191182336*y^6 - 1811939328*y^5 - 2415919104*y^4 - 2147483648*y^3 - 12884901888*y^2 + 68719476736, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 3*x^34 - 2*x^33 - 9*x^32 - 27*x^31 + 71*x^30 - 6*x^29 + 524*x^27 + 1812*x^26 - 1104*x^25 - 2280*x^24 - 15024*x^23 - 21696*x^22 - 3264*x^21 + 97152*x^20 + 16896*x^19 + 301568*x^18 + 67584*x^17 + 1554432*x^16 - 208896*x^15 - 5554176*x^14 - 15384576*x^13 - 9338880*x^12 - 18087936*x^11 + 118751232*x^10 + 137363456*x^9 - 25165824*x^7 + 1191182336*x^6 - 1811939328*x^5 - 2415919104*x^4 - 2147483648*x^3 - 12884901888*x^2 + 68719476736);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^36 - 3*x^34 - 2*x^33 - 9*x^32 - 27*x^31 + 71*x^30 - 6*x^29 + 524*x^27 + 1812*x^26 - 1104*x^25 - 2280*x^24 - 15024*x^23 - 21696*x^22 - 3264*x^21 + 97152*x^20 + 16896*x^19 + 301568*x^18 + 67584*x^17 + 1554432*x^16 - 208896*x^15 - 5554176*x^14 - 15384576*x^13 - 9338880*x^12 - 18087936*x^11 + 118751232*x^10 + 137363456*x^9 - 25165824*x^7 + 1191182336*x^6 - 1811939328*x^5 - 2415919104*x^4 - 2147483648*x^3 - 12884901888*x^2 + 68719476736)
 

\( x^{36} - 3 x^{34} - 2 x^{33} - 9 x^{32} - 27 x^{31} + 71 x^{30} - 6 x^{29} + 524 x^{27} + \cdots + 68719476736 \) 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:   \(17303778994656659820211322036533666992057745030749670455635346022689551189016576\) \(\medspace = 2^{18}\cdot 3^{48}\cdot 7^{18}\cdot 67^{12}\cdot 199^{6}\) 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:  \(158.88\)
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:  $2^{3/2}3^{4/3}7^{3/4}67^{1/2}199^{1/2}\approx 6081.267901071334$
Ramified primes:   \(2\), \(3\), \(7\), \(67\), \(199\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{1}{8}a^{8}+\frac{1}{8}a^{7}-\frac{3}{8}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{13}+\frac{1}{16}a^{11}-\frac{1}{8}a^{10}+\frac{3}{16}a^{9}-\frac{3}{16}a^{8}-\frac{5}{16}a^{7}-\frac{1}{8}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{32}a^{14}+\frac{1}{32}a^{12}-\frac{1}{16}a^{11}+\frac{3}{32}a^{10}-\frac{3}{32}a^{9}-\frac{5}{32}a^{8}-\frac{1}{16}a^{7}+\frac{1}{8}a^{6}+\frac{3}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{15}+\frac{1}{64}a^{13}-\frac{1}{32}a^{12}-\frac{5}{64}a^{11}-\frac{3}{64}a^{10}-\frac{13}{64}a^{9}+\frac{7}{32}a^{8}+\frac{3}{16}a^{7}-\frac{7}{16}a^{6}-\frac{7}{16}a^{5}-\frac{1}{2}a^{4}-\frac{3}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{128}a^{16}+\frac{1}{128}a^{14}-\frac{1}{64}a^{13}-\frac{5}{128}a^{12}-\frac{3}{128}a^{11}-\frac{13}{128}a^{10}+\frac{7}{64}a^{9}+\frac{3}{32}a^{8}+\frac{9}{32}a^{7}-\frac{7}{32}a^{6}-\frac{1}{4}a^{5}-\frac{3}{16}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{256}a^{17}+\frac{1}{256}a^{15}-\frac{1}{128}a^{14}-\frac{5}{256}a^{13}-\frac{3}{256}a^{12}-\frac{13}{256}a^{11}+\frac{7}{128}a^{10}-\frac{13}{64}a^{9}+\frac{9}{64}a^{8}-\frac{23}{64}a^{7}+\frac{3}{8}a^{6}+\frac{5}{32}a^{5}+\frac{5}{16}a^{4}-\frac{1}{8}a^{3}$, $\frac{1}{512}a^{18}+\frac{1}{512}a^{16}-\frac{1}{256}a^{15}-\frac{5}{512}a^{14}-\frac{3}{512}a^{13}-\frac{13}{512}a^{12}+\frac{7}{256}a^{11}-\frac{13}{128}a^{10}+\frac{9}{128}a^{9}-\frac{23}{128}a^{8}+\frac{3}{16}a^{7}+\frac{5}{64}a^{6}-\frac{11}{32}a^{5}+\frac{7}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{1024}a^{19}+\frac{1}{1024}a^{17}-\frac{1}{512}a^{16}-\frac{5}{1024}a^{15}-\frac{3}{1024}a^{14}-\frac{13}{1024}a^{13}+\frac{7}{512}a^{12}+\frac{19}{256}a^{11}+\frac{9}{256}a^{10}+\frac{9}{256}a^{9}-\frac{5}{32}a^{8}+\frac{53}{128}a^{7}-\frac{3}{64}a^{6}-\frac{13}{32}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{2048}a^{20}+\frac{1}{2048}a^{18}-\frac{1}{1024}a^{17}-\frac{5}{2048}a^{16}-\frac{3}{2048}a^{15}-\frac{13}{2048}a^{14}+\frac{7}{1024}a^{13}+\frac{19}{512}a^{12}+\frac{9}{512}a^{11}+\frac{9}{512}a^{10}-\frac{5}{64}a^{9}+\frac{53}{256}a^{8}+\frac{61}{128}a^{7}-\frac{13}{64}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4096}a^{21}+\frac{1}{4096}a^{19}-\frac{1}{2048}a^{18}-\frac{5}{4096}a^{17}-\frac{3}{4096}a^{16}-\frac{13}{4096}a^{15}+\frac{7}{2048}a^{14}+\frac{19}{1024}a^{13}+\frac{9}{1024}a^{12}-\frac{119}{1024}a^{11}-\frac{5}{128}a^{10}-\frac{11}{512}a^{9}-\frac{3}{256}a^{8}-\frac{61}{128}a^{7}+\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{5}{16}a^{4}-\frac{1}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{8192}a^{22}+\frac{1}{8192}a^{20}-\frac{1}{4096}a^{19}-\frac{5}{8192}a^{18}-\frac{3}{8192}a^{17}-\frac{13}{8192}a^{16}+\frac{7}{4096}a^{15}+\frac{19}{2048}a^{14}+\frac{9}{2048}a^{13}-\frac{119}{2048}a^{12}-\frac{5}{256}a^{11}-\frac{11}{1024}a^{10}-\frac{3}{512}a^{9}-\frac{61}{256}a^{8}-\frac{7}{16}a^{7}-\frac{1}{8}a^{6}+\frac{11}{32}a^{5}+\frac{7}{16}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16384}a^{23}+\frac{1}{16384}a^{21}-\frac{1}{8192}a^{20}-\frac{5}{16384}a^{19}-\frac{3}{16384}a^{18}-\frac{13}{16384}a^{17}+\frac{7}{8192}a^{16}+\frac{19}{4096}a^{15}+\frac{9}{4096}a^{14}-\frac{119}{4096}a^{13}-\frac{5}{512}a^{12}+\frac{245}{2048}a^{11}-\frac{3}{1024}a^{10}+\frac{3}{512}a^{9}+\frac{1}{32}a^{8}+\frac{5}{16}a^{7}+\frac{19}{64}a^{6}-\frac{13}{32}a^{5}+\frac{1}{16}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{32768}a^{24}+\frac{1}{32768}a^{22}-\frac{1}{16384}a^{21}-\frac{5}{32768}a^{20}-\frac{3}{32768}a^{19}-\frac{13}{32768}a^{18}+\frac{7}{16384}a^{17}+\frac{19}{8192}a^{16}+\frac{9}{8192}a^{15}-\frac{119}{8192}a^{14}-\frac{5}{1024}a^{13}+\frac{245}{4096}a^{12}+\frac{253}{2048}a^{11}+\frac{3}{1024}a^{10}-\frac{7}{64}a^{9}-\frac{3}{32}a^{8}+\frac{35}{128}a^{7}-\frac{5}{64}a^{6}+\frac{5}{32}a^{5}+\frac{1}{8}a^{4}$, $\frac{1}{65536}a^{25}+\frac{1}{65536}a^{23}-\frac{1}{32768}a^{22}-\frac{5}{65536}a^{21}-\frac{3}{65536}a^{20}-\frac{13}{65536}a^{19}+\frac{7}{32768}a^{18}+\frac{19}{16384}a^{17}+\frac{9}{16384}a^{16}-\frac{119}{16384}a^{15}-\frac{5}{2048}a^{14}+\frac{245}{8192}a^{13}+\frac{253}{4096}a^{12}+\frac{3}{2048}a^{11}-\frac{7}{128}a^{10}+\frac{13}{64}a^{9}+\frac{35}{256}a^{8}-\frac{37}{128}a^{7}+\frac{5}{64}a^{6}-\frac{3}{16}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{262144}a^{26}-\frac{3}{262144}a^{24}-\frac{1}{131072}a^{23}-\frac{9}{262144}a^{22}-\frac{27}{262144}a^{21}-\frac{57}{262144}a^{20}-\frac{3}{131072}a^{19}-\frac{1}{2048}a^{18}-\frac{61}{65536}a^{17}+\frac{101}{65536}a^{16}-\frac{109}{16384}a^{15}-\frac{77}{32768}a^{14}-\frac{475}{16384}a^{13}+\frac{229}{4096}a^{12}+\frac{181}{4096}a^{11}+\frac{51}{2048}a^{10}-\frac{7}{512}a^{9}+\frac{43}{512}a^{8}+\frac{31}{64}a^{7}+\frac{45}{128}a^{6}+\frac{3}{64}a^{5}-\frac{1}{16}a^{4}+\frac{7}{16}a^{3}+\frac{3}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{524288}a^{27}-\frac{3}{524288}a^{25}-\frac{1}{262144}a^{24}-\frac{9}{524288}a^{23}-\frac{27}{524288}a^{22}-\frac{57}{524288}a^{21}-\frac{3}{262144}a^{20}-\frac{1}{4096}a^{19}-\frac{61}{131072}a^{18}+\frac{101}{131072}a^{17}-\frac{109}{32768}a^{16}-\frac{77}{65536}a^{15}-\frac{475}{32768}a^{14}+\frac{229}{8192}a^{13}+\frac{181}{8192}a^{12}-\frac{461}{4096}a^{11}-\frac{7}{1024}a^{10}-\frac{85}{1024}a^{9}-\frac{1}{128}a^{8}+\frac{77}{256}a^{7}+\frac{51}{128}a^{6}+\frac{3}{32}a^{5}+\frac{15}{32}a^{4}+\frac{3}{16}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{2097152}a^{28}-\frac{3}{2097152}a^{26}-\frac{1}{1048576}a^{25}-\frac{9}{2097152}a^{24}-\frac{27}{2097152}a^{23}+\frac{71}{2097152}a^{22}-\frac{3}{1048576}a^{21}+\frac{131}{524288}a^{19}+\frac{453}{524288}a^{18}-\frac{69}{131072}a^{17}-\frac{285}{262144}a^{16}-\frac{939}{131072}a^{15}-\frac{339}{32768}a^{14}-\frac{51}{32768}a^{13}+\frac{759}{16384}a^{12}+\frac{33}{4096}a^{11}-\frac{435}{4096}a^{10}+\frac{33}{1024}a^{9}-\frac{9}{1024}a^{8}+\frac{205}{512}a^{7}+\frac{13}{128}a^{6}+\frac{53}{128}a^{5}+\frac{19}{64}a^{4}+\frac{3}{8}a^{3}+\frac{1}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{4194304}a^{29}-\frac{3}{4194304}a^{27}-\frac{1}{2097152}a^{26}-\frac{9}{4194304}a^{25}-\frac{27}{4194304}a^{24}+\frac{71}{4194304}a^{23}-\frac{3}{2097152}a^{22}+\frac{131}{1048576}a^{20}+\frac{453}{1048576}a^{19}-\frac{69}{262144}a^{18}-\frac{285}{524288}a^{17}-\frac{939}{262144}a^{16}-\frac{339}{65536}a^{15}-\frac{51}{65536}a^{14}+\frac{759}{32768}a^{13}+\frac{33}{8192}a^{12}+\frac{589}{8192}a^{11}+\frac{33}{2048}a^{10}-\frac{265}{2048}a^{9}-\frac{51}{1024}a^{8}+\frac{45}{256}a^{7}+\frac{85}{256}a^{6}+\frac{35}{128}a^{5}+\frac{3}{16}a^{4}-\frac{3}{16}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{83886080}a^{30}-\frac{1}{10485760}a^{29}-\frac{3}{83886080}a^{28}-\frac{21}{41943040}a^{27}+\frac{71}{83886080}a^{26}-\frac{19}{83886080}a^{25}-\frac{801}{83886080}a^{24}-\frac{243}{8388608}a^{23}-\frac{17}{1048576}a^{22}+\frac{931}{20971520}a^{21}+\frac{2621}{20971520}a^{20}-\frac{2263}{5242880}a^{19}+\frac{10131}{10485760}a^{18}+\frac{1641}{5242880}a^{17}-\frac{2441}{1310720}a^{16}-\frac{3731}{1310720}a^{15}+\frac{8903}{655360}a^{14}-\frac{607}{163840}a^{13}-\frac{579}{163840}a^{12}-\frac{1671}{40960}a^{11}-\frac{653}{40960}a^{10}-\frac{4231}{20480}a^{9}+\frac{57}{1024}a^{8}+\frac{377}{1024}a^{7}-\frac{133}{2560}a^{6}-\frac{71}{320}a^{5}+\frac{1}{320}a^{4}+\frac{1}{10}a^{3}+\frac{9}{40}a^{2}-\frac{1}{10}a-\frac{1}{5}$, $\frac{1}{24494735360}a^{31}-\frac{5}{1224736768}a^{30}+\frac{253}{24494735360}a^{29}+\frac{2277}{12247367680}a^{28}+\frac{2163}{4898947072}a^{27}+\frac{41609}{24494735360}a^{26}-\frac{86253}{24494735360}a^{25}-\frac{70449}{12247367680}a^{24}+\frac{6899}{612368384}a^{23}-\frac{201989}{6123683840}a^{22}+\frac{436809}{6123683840}a^{21}-\frac{1713}{765460480}a^{20}-\frac{1389957}{3061841920}a^{19}+\frac{288059}{306184192}a^{18}-\frac{31349}{23920640}a^{17}-\frac{542139}{382730240}a^{16}+\frac{1457769}{191365120}a^{15}-\frac{108953}{23920640}a^{14}+\frac{72417}{9568256}a^{13}+\frac{271423}{5980160}a^{12}+\frac{185659}{11960320}a^{11}+\frac{105147}{5980160}a^{10}-\frac{1087}{5840}a^{9}-\frac{27951}{299008}a^{8}+\frac{357157}{747520}a^{7}+\frac{37727}{186880}a^{6}+\frac{15103}{93440}a^{5}-\frac{873}{4672}a^{4}-\frac{287}{5840}a^{3}+\frac{617}{2920}a^{2}+\frac{17}{292}a+\frac{178}{365}$, $\frac{1}{76\cdots 80}a^{32}-\frac{29481}{19\cdots 20}a^{31}-\frac{25558307}{76\cdots 80}a^{30}-\frac{15760939}{38\cdots 40}a^{29}+\frac{1203935519}{76\cdots 80}a^{28}+\frac{5376402089}{76\cdots 80}a^{27}+\frac{8651909811}{76\cdots 80}a^{26}-\frac{19288995313}{38\cdots 40}a^{25}+\frac{2985154439}{960524305039360}a^{24}+\frac{34466494691}{19\cdots 20}a^{23}+\frac{6179678981}{384209722015744}a^{22}-\frac{1440432425}{48026215251968}a^{21}-\frac{224061379429}{960524305039360}a^{20}+\frac{221919009359}{480262152519680}a^{19}-\frac{20021488601}{30016384532480}a^{18}+\frac{12200264537}{120065538129920}a^{17}+\frac{1741323109}{822366699520}a^{16}+\frac{8821512273}{1500819226624}a^{15}+\frac{71756415369}{15008192266240}a^{14}+\frac{20285183949}{938012016640}a^{13}-\frac{30202392393}{750409613312}a^{12}+\frac{60506078039}{1876024033280}a^{11}-\frac{3105172469}{29312875520}a^{10}+\frac{27074273393}{469006008320}a^{9}+\frac{36266562937}{234503004160}a^{8}+\frac{4692873927}{11725150208}a^{7}-\frac{13774369191}{29312875520}a^{6}-\frac{703890409}{3664109440}a^{5}+\frac{259206519}{916027360}a^{4}-\frac{205510959}{916027360}a^{3}+\frac{124292261}{458013680}a^{2}-\frac{9335081}{57251710}a+\frac{7685862}{28625855}$, $\frac{1}{30\cdots 20}a^{33}-\frac{483219}{30\cdots 20}a^{31}+\frac{53515743}{15\cdots 60}a^{30}+\frac{1329801383}{30\cdots 20}a^{29}-\frac{1079006527}{61\cdots 04}a^{28}-\frac{19724359977}{30\cdots 20}a^{27}-\frac{16443130827}{15\cdots 60}a^{26}+\frac{2637009213}{19\cdots 20}a^{25}-\frac{12343452053}{76\cdots 80}a^{24}-\frac{87448278491}{76\cdots 80}a^{23}+\frac{67923772591}{19\cdots 20}a^{22}+\frac{56379497375}{768419444031488}a^{21}+\frac{65134194449}{384209722015744}a^{20}+\frac{13947100411}{96052430503936}a^{19}+\frac{58126259821}{96052430503936}a^{18}-\frac{145617172793}{240131076259840}a^{17}+\frac{190995549271}{60032769064960}a^{16}-\frac{328809433031}{60032769064960}a^{15}+\frac{41398380029}{3001638453248}a^{14}+\frac{276770473443}{15008192266240}a^{13}-\frac{70911704727}{1500819226624}a^{12}-\frac{91688840777}{1876024033280}a^{11}-\frac{52313690183}{1876024033280}a^{10}+\frac{4926565047}{187602403328}a^{9}+\frac{1983069843}{58625751040}a^{8}+\frac{10351017319}{23450300416}a^{7}+\frac{1715071769}{29312875520}a^{6}-\frac{249122995}{1465643776}a^{5}-\frac{633479197}{3664109440}a^{4}-\frac{314152553}{1832054720}a^{3}-\frac{20112001}{458013680}a^{2}-\frac{20512543}{57251710}a-\frac{312574}{28625855}$, $\frac{1}{68\cdots 80}a^{34}+\frac{145907}{17\cdots 20}a^{33}+\frac{1126109}{68\cdots 80}a^{32}-\frac{446998841363}{34\cdots 40}a^{31}+\frac{155900913129919}{68\cdots 80}a^{30}-\frac{55\cdots 31}{68\cdots 80}a^{29}+\frac{18\cdots 51}{68\cdots 80}a^{28}+\frac{10\cdots 67}{34\cdots 40}a^{27}-\frac{26\cdots 73}{86\cdots 60}a^{26}-\frac{38\cdots 57}{17\cdots 20}a^{25}+\frac{10\cdots 69}{17\cdots 20}a^{24}+\frac{64\cdots 93}{21\cdots 40}a^{23}+\frac{30\cdots 27}{86\cdots 60}a^{22}+\frac{23\cdots 43}{43\cdots 80}a^{21}+\frac{305475841379023}{16\cdots 80}a^{20}-\frac{12\cdots 23}{10\cdots 20}a^{19}-\frac{43\cdots 63}{10\cdots 92}a^{18}+\frac{28\cdots 09}{67\cdots 20}a^{17}-\frac{76\cdots 03}{13\cdots 40}a^{16}+\frac{55\cdots 11}{84\cdots 40}a^{15}+\frac{77\cdots 63}{67\cdots 12}a^{14}+\frac{20\cdots 63}{33\cdots 56}a^{13}-\frac{13\cdots 11}{26\cdots 20}a^{12}+\frac{31\cdots 17}{42\cdots 20}a^{11}+\frac{95\cdots 53}{21\cdots 60}a^{10}-\frac{19\cdots 77}{10\cdots 08}a^{9}+\frac{31\cdots 45}{52\cdots 04}a^{8}+\frac{670838446172647}{82\cdots 60}a^{7}-\frac{74\cdots 97}{16\cdots 20}a^{6}+\frac{37\cdots 31}{82\cdots 60}a^{5}+\frac{20\cdots 01}{41\cdots 80}a^{4}+\frac{203970991870569}{513535014316960}a^{3}+\frac{6677033842149}{16047969197405}a^{2}-\frac{1903627408601}{6419187678962}a-\frac{4853594006142}{16047969197405}$, $\frac{1}{46\cdots 40}a^{35}-\frac{1}{14\cdots 20}a^{34}+\frac{942181}{64\cdots 80}a^{33}+\frac{1245823}{13\cdots 60}a^{32}-\frac{53490117455673}{46\cdots 40}a^{31}+\frac{44\cdots 17}{46\cdots 40}a^{30}-\frac{28\cdots 53}{46\cdots 40}a^{29}-\frac{53\cdots 35}{46\cdots 84}a^{28}+\frac{25\cdots 47}{58\cdots 48}a^{27}-\frac{15\cdots 73}{11\cdots 60}a^{26}-\frac{61\cdots 67}{11\cdots 60}a^{25}-\frac{42\cdots 49}{29\cdots 40}a^{24}-\frac{12\cdots 21}{58\cdots 80}a^{23}-\frac{17\cdots 39}{29\cdots 40}a^{22}-\frac{19\cdots 89}{73\cdots 60}a^{21}+\frac{16\cdots 69}{73\cdots 60}a^{20}-\frac{12\cdots 93}{36\cdots 80}a^{19}+\frac{81\cdots 79}{91\cdots 20}a^{18}+\frac{16\cdots 93}{91\cdots 20}a^{17}+\frac{70\cdots 09}{22\cdots 80}a^{16}+\frac{99\cdots 43}{22\cdots 80}a^{15}-\frac{73\cdots 83}{11\cdots 40}a^{14}-\frac{37\cdots 01}{28\cdots 60}a^{13}+\frac{42\cdots 69}{28\cdots 60}a^{12}+\frac{65\cdots 27}{14\cdots 80}a^{11}+\frac{45\cdots 51}{44\cdots 40}a^{10}-\frac{32\cdots 93}{17\cdots 60}a^{9}-\frac{47\cdots 13}{89\cdots 68}a^{8}-\frac{51\cdots 13}{11\cdots 60}a^{7}+\frac{98\cdots 67}{55\cdots 80}a^{6}-\frac{12\cdots 21}{27\cdots 40}a^{5}+\frac{33\cdots 31}{69\cdots 60}a^{4}-\frac{77785234168793}{513535014316960}a^{3}-\frac{169592888391153}{43\cdots 60}a^{2}+\frac{71626688398991}{218252381084708}a+\frac{41978613324161}{272815476355885}$ 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:  Not computed
Index:  $1$
Inessential primes:  None

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:   \( -1 \)  (order $2$) 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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot R \cdot h}{2\cdot\sqrt{17303778994656659820211322036533666992057745030749670455635346022689551189016576}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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 - 3*x^34 - 2*x^33 - 9*x^32 - 27*x^31 + 71*x^30 - 6*x^29 + 524*x^27 + 1812*x^26 - 1104*x^25 - 2280*x^24 - 15024*x^23 - 21696*x^22 - 3264*x^21 + 97152*x^20 + 16896*x^19 + 301568*x^18 + 67584*x^17 + 1554432*x^16 - 208896*x^15 - 5554176*x^14 - 15384576*x^13 - 9338880*x^12 - 18087936*x^11 + 118751232*x^10 + 137363456*x^9 - 25165824*x^7 + 1191182336*x^6 - 1811939328*x^5 - 2415919104*x^4 - 2147483648*x^3 - 12884901888*x^2 + 68719476736) 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 - 3*x^34 - 2*x^33 - 9*x^32 - 27*x^31 + 71*x^30 - 6*x^29 + 524*x^27 + 1812*x^26 - 1104*x^25 - 2280*x^24 - 15024*x^23 - 21696*x^22 - 3264*x^21 + 97152*x^20 + 16896*x^19 + 301568*x^18 + 67584*x^17 + 1554432*x^16 - 208896*x^15 - 5554176*x^14 - 15384576*x^13 - 9338880*x^12 - 18087936*x^11 + 118751232*x^10 + 137363456*x^9 - 25165824*x^7 + 1191182336*x^6 - 1811939328*x^5 - 2415919104*x^4 - 2147483648*x^3 - 12884901888*x^2 + 68719476736, 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 - 3*x^34 - 2*x^33 - 9*x^32 - 27*x^31 + 71*x^30 - 6*x^29 + 524*x^27 + 1812*x^26 - 1104*x^25 - 2280*x^24 - 15024*x^23 - 21696*x^22 - 3264*x^21 + 97152*x^20 + 16896*x^19 + 301568*x^18 + 67584*x^17 + 1554432*x^16 - 208896*x^15 - 5554176*x^14 - 15384576*x^13 - 9338880*x^12 - 18087936*x^11 + 118751232*x^10 + 137363456*x^9 - 25165824*x^7 + 1191182336*x^6 - 1811939328*x^5 - 2415919104*x^4 - 2147483648*x^3 - 12884901888*x^2 + 68719476736); 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 - 3*x^34 - 2*x^33 - 9*x^32 - 27*x^31 + 71*x^30 - 6*x^29 + 524*x^27 + 1812*x^26 - 1104*x^25 - 2280*x^24 - 15024*x^23 - 21696*x^22 - 3264*x^21 + 97152*x^20 + 16896*x^19 + 301568*x^18 + 67584*x^17 + 1554432*x^16 - 208896*x^15 - 5554176*x^14 - 15384576*x^13 - 9338880*x^12 - 18087936*x^11 + 118751232*x^10 + 137363456*x^9 - 25165824*x^7 + 1191182336*x^6 - 1811939328*x^5 - 2415919104*x^4 - 2147483648*x^3 - 12884901888*x^2 + 68719476736); 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

$A_4^2:C_2^3$ (as 36T1803):

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 1152
The 80 conjugacy class representatives for $A_4^2:C_2^3$
Character table for $A_4^2:C_2^3$

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.469.1, 6.0.1305639.1, 6.0.12317816.1, 9.9.54824361792669.2, 18.18.4159781123407415509472868801127860417024.1, 18.0.527850880803275932384437883608576.1, 18.0.23686800310947837950260049204673039.1

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: deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, some data not computed
Minimal sibling: not computed

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 ${\href{/padicField/5.3.0.1}{3} }^{12}$ R ${\href{/padicField/11.6.0.1}{6} }^{6}$ ${\href{/padicField/13.3.0.1}{3} }^{12}$ ${\href{/padicField/17.4.0.1}{4} }^{6}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{6}{,}\,{\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.3.0.1}{3} }^{12}$ ${\href{/padicField/29.6.0.1}{6} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{6}$ ${\href{/padicField/43.12.0.1}{12} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.12.0.1}{12} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{6}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{6}$

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 2.3.2.9a1.5$x^{6} + 2 x^{4} + 6 x^{3} + x^{2} + 6 x + 7$$2$$3$$9$$C_6$$$[3]^{3}$$
2.3.2.9a1.5$x^{6} + 2 x^{4} + 6 x^{3} + x^{2} + 6 x + 7$$2$$3$$9$$C_6$$$[3]^{3}$$
2.6.1.0a1.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$$[\ ]^{6}$$
2.6.1.0a1.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$$[\ ]^{6}$$
2.6.1.0a1.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$$[\ ]^{6}$$
2.6.1.0a1.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(3\) Copy content Toggle raw display 3.6.3.24a2.1$x^{18} + 6 x^{16} + 15 x^{14} + 6 x^{13} + 32 x^{12} + 24 x^{11} + 63 x^{10} + 36 x^{9} + 90 x^{8} + 72 x^{7} + 109 x^{6} + 102 x^{5} + 96 x^{4} + 56 x^{3} + 84 x^{2} + 72 x + 35$$3$$6$$24$$C_6 \times C_3$$$[2]^{6}$$
3.6.3.24a2.1$x^{18} + 6 x^{16} + 15 x^{14} + 6 x^{13} + 32 x^{12} + 24 x^{11} + 63 x^{10} + 36 x^{9} + 90 x^{8} + 72 x^{7} + 109 x^{6} + 102 x^{5} + 96 x^{4} + 56 x^{3} + 84 x^{2} + 72 x + 35$$3$$6$$24$$C_6 \times C_3$$$[2]^{6}$$
\(7\) 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$$$[\ ]_{4}^{6}$$
\(67\) Copy content Toggle raw display 67.6.1.0a1.1$x^{6} + 63 x^{3} + 49 x^{2} + 55 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
67.6.1.0a1.1$x^{6} + 63 x^{3} + 49 x^{2} + 55 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
67.6.2.6a1.2$x^{12} + 126 x^{9} + 98 x^{8} + 110 x^{7} + 3973 x^{6} + 6174 x^{5} + 9331 x^{4} + 5642 x^{3} + 3221 x^{2} + 220 x + 71$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
67.6.2.6a1.2$x^{12} + 126 x^{9} + 98 x^{8} + 110 x^{7} + 3973 x^{6} + 6174 x^{5} + 9331 x^{4} + 5642 x^{3} + 3221 x^{2} + 220 x + 71$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
\(199\) Copy content Toggle raw display 199.2.1.0a1.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
199.2.1.0a1.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
199.2.1.0a1.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
199.2.1.0a1.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
199.2.1.0a1.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
199.2.1.0a1.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
199.2.1.0a1.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
199.2.1.0a1.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
199.2.1.0a1.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
199.2.1.0a1.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
199.2.1.0a1.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
199.2.1.0a1.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
199.2.2.2a1.2$x^{4} + 386 x^{3} + 37255 x^{2} + 1158 x + 208$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
199.2.2.2a1.2$x^{4} + 386 x^{3} + 37255 x^{2} + 1158 x + 208$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
199.2.2.2a1.2$x^{4} + 386 x^{3} + 37255 x^{2} + 1158 x + 208$$2$$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)