↑

Properties

Label 32.0.681...000.2
Degree $32$
Signature $[0, 16]$
Discriminant $6.815\times 10^{55}$
Root discriminant \(55.56\)
Ramified primes $2,3,5,29,521$
Class number not computed
Class group not computed
Galois group $D_4^2:C_2^3$ (as 32T12882)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 35*x^30 + 54*x^29 + 582*x^28 - 724*x^27 - 5801*x^26 + 6656*x^25 + 38405*x^24 - 49984*x^23 - 177937*x^22 + 300832*x^21 + 590114*x^20 - 1365710*x^19 - 1185291*x^18 + 4579318*x^17 + 154147*x^16 - 11942188*x^15 + 7662420*x^14 + 16564120*x^13 - 23303048*x^12 - 2384240*x^11 + 68187376*x^10 - 79122656*x^9 + 115265728*x^8 - 58417728*x^7 + 58682560*x^6 - 14078848*x^5 + 11490944*x^4 - 313088*x^3 + 444416*x^2 + 256)
 
gp: K = bnfinit(y^32 - 2*y^31 - 35*y^30 + 54*y^29 + 582*y^28 - 724*y^27 - 5801*y^26 + 6656*y^25 + 38405*y^24 - 49984*y^23 - 177937*y^22 + 300832*y^21 + 590114*y^20 - 1365710*y^19 - 1185291*y^18 + 4579318*y^17 + 154147*y^16 - 11942188*y^15 + 7662420*y^14 + 16564120*y^13 - 23303048*y^12 - 2384240*y^11 + 68187376*y^10 - 79122656*y^9 + 115265728*y^8 - 58417728*y^7 + 58682560*y^6 - 14078848*y^5 + 11490944*y^4 - 313088*y^3 + 444416*y^2 + 256, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 2*x^31 - 35*x^30 + 54*x^29 + 582*x^28 - 724*x^27 - 5801*x^26 + 6656*x^25 + 38405*x^24 - 49984*x^23 - 177937*x^22 + 300832*x^21 + 590114*x^20 - 1365710*x^19 - 1185291*x^18 + 4579318*x^17 + 154147*x^16 - 11942188*x^15 + 7662420*x^14 + 16564120*x^13 - 23303048*x^12 - 2384240*x^11 + 68187376*x^10 - 79122656*x^9 + 115265728*x^8 - 58417728*x^7 + 58682560*x^6 - 14078848*x^5 + 11490944*x^4 - 313088*x^3 + 444416*x^2 + 256);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 2*x^31 - 35*x^30 + 54*x^29 + 582*x^28 - 724*x^27 - 5801*x^26 + 6656*x^25 + 38405*x^24 - 49984*x^23 - 177937*x^22 + 300832*x^21 + 590114*x^20 - 1365710*x^19 - 1185291*x^18 + 4579318*x^17 + 154147*x^16 - 11942188*x^15 + 7662420*x^14 + 16564120*x^13 - 23303048*x^12 - 2384240*x^11 + 68187376*x^10 - 79122656*x^9 + 115265728*x^8 - 58417728*x^7 + 58682560*x^6 - 14078848*x^5 + 11490944*x^4 - 313088*x^3 + 444416*x^2 + 256)
 

\( x^{32} - 2 x^{31} - 35 x^{30} + 54 x^{29} + 582 x^{28} - 724 x^{27} - 5801 x^{26} + 6656 x^{25} + \cdots + 256 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(68145128239473945155058695587402151362560000000000000000\) \(\medspace = 2^{48}\cdot 3^{16}\cdot 5^{16}\cdot 29^{8}\cdot 521^{4}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(55.56\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}3^{1/2}5^{1/2}29^{1/2}521^{1/2}\approx 1346.5065911461406$
Ramified primes:   \(2\), \(3\), \(5\), \(29\), \(521\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{20}a^{18}+\frac{3}{10}a^{17}+\frac{7}{20}a^{16}-\frac{1}{2}a^{15}-\frac{1}{5}a^{14}-\frac{2}{5}a^{13}-\frac{1}{4}a^{12}+\frac{3}{20}a^{10}+\frac{9}{20}a^{8}+\frac{2}{5}a^{7}+\frac{2}{5}a^{6}+\frac{3}{10}a^{5}+\frac{1}{20}a^{4}-\frac{1}{10}a^{3}+\frac{1}{20}a^{2}-\frac{2}{5}a+\frac{3}{10}$, $\frac{1}{20}a^{19}-\frac{9}{20}a^{17}+\frac{2}{5}a^{16}-\frac{1}{5}a^{15}-\frac{1}{5}a^{14}+\frac{3}{20}a^{13}-\frac{1}{2}a^{12}+\frac{3}{20}a^{11}+\frac{1}{10}a^{10}+\frac{9}{20}a^{9}-\frac{3}{10}a^{8}-\frac{1}{10}a^{6}+\frac{1}{4}a^{5}-\frac{2}{5}a^{4}-\frac{7}{20}a^{3}+\frac{3}{10}a^{2}-\frac{3}{10}a+\frac{1}{5}$, $\frac{1}{80}a^{20}-\frac{1}{40}a^{19}+\frac{1}{80}a^{18}-\frac{17}{40}a^{17}+\frac{3}{8}a^{16}-\frac{9}{20}a^{15}-\frac{29}{80}a^{14}+\frac{3}{10}a^{13}-\frac{7}{80}a^{12}+\frac{1}{5}a^{11}+\frac{3}{16}a^{10}+\frac{1}{5}a^{9}+\frac{1}{40}a^{8}+\frac{9}{40}a^{7}+\frac{9}{80}a^{6}-\frac{9}{40}a^{5}-\frac{1}{80}a^{4}+\frac{1}{4}a^{3}-\frac{7}{20}a^{2}+\frac{1}{5}a-\frac{7}{20}$, $\frac{1}{80}a^{21}+\frac{1}{80}a^{19}+\frac{19}{40}a^{17}-\frac{1}{2}a^{16}-\frac{37}{80}a^{15}-\frac{9}{40}a^{14}+\frac{37}{80}a^{13}-\frac{19}{40}a^{12}-\frac{21}{80}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{17}{40}a^{8}-\frac{19}{80}a^{7}+\frac{1}{10}a^{6}+\frac{3}{16}a^{5}+\frac{9}{40}a^{4}+\frac{1}{5}a^{2}-\frac{9}{20}a-\frac{1}{10}$, $\frac{1}{320}a^{22}-\frac{1}{160}a^{21}-\frac{1}{320}a^{20}+\frac{1}{160}a^{19}-\frac{1}{80}a^{18}-\frac{3}{20}a^{17}-\frac{57}{320}a^{16}-\frac{7}{20}a^{15}+\frac{51}{320}a^{14}+\frac{1}{4}a^{13}+\frac{109}{320}a^{12}-\frac{1}{4}a^{11}+\frac{1}{16}a^{10}-\frac{63}{160}a^{9}-\frac{15}{64}a^{8}+\frac{9}{32}a^{7}-\frac{19}{320}a^{6}-\frac{7}{40}a^{5}+\frac{43}{160}a^{4}-\frac{3}{40}a^{3}+\frac{27}{80}a^{2}+\frac{1}{10}a+\frac{19}{40}$, $\frac{1}{320}a^{23}-\frac{1}{320}a^{21}+\frac{1}{80}a^{19}-\frac{1}{40}a^{18}-\frac{33}{320}a^{17}-\frac{5}{32}a^{16}+\frac{159}{320}a^{15}-\frac{41}{160}a^{14}+\frac{33}{320}a^{13}+\frac{33}{160}a^{12}+\frac{3}{10}a^{11}+\frac{9}{160}a^{10}-\frac{47}{320}a^{9}-\frac{21}{80}a^{8}+\frac{149}{320}a^{7}+\frac{1}{160}a^{6}+\frac{1}{160}a^{5}-\frac{13}{80}a^{4}-\frac{9}{80}a^{3}+\frac{1}{8}a^{2}+\frac{1}{40}a-\frac{1}{4}$, $\frac{1}{1280}a^{24}-\frac{1}{640}a^{23}+\frac{1}{1280}a^{22}-\frac{1}{640}a^{21}-\frac{3}{640}a^{20}+\frac{1}{64}a^{19}-\frac{1}{1280}a^{18}-\frac{5}{16}a^{17}-\frac{63}{1280}a^{16}-\frac{5}{16}a^{15}-\frac{621}{1280}a^{14}-\frac{3}{16}a^{13}-\frac{41}{640}a^{12}-\frac{47}{640}a^{11}+\frac{57}{256}a^{10}-\frac{273}{640}a^{9}+\frac{23}{1280}a^{8}+\frac{159}{320}a^{7}+\frac{7}{80}a^{6}+\frac{31}{160}a^{5}+\frac{59}{160}a^{4}-\frac{37}{80}a^{3}+\frac{31}{80}a^{2}+\frac{7}{20}a+\frac{19}{80}$, $\frac{1}{1280}a^{25}+\frac{1}{1280}a^{23}+\frac{1}{640}a^{21}-\frac{1}{160}a^{20}-\frac{5}{256}a^{19}-\frac{1}{640}a^{18}+\frac{157}{1280}a^{17}+\frac{133}{640}a^{16}+\frac{351}{1280}a^{15}-\frac{177}{640}a^{14}-\frac{35}{128}a^{13}-\frac{17}{128}a^{12}-\frac{99}{256}a^{11}-\frac{31}{80}a^{10}-\frac{53}{256}a^{9}-\frac{55}{128}a^{8}-\frac{37}{320}a^{7}+\frac{29}{80}a^{6}-\frac{9}{40}a^{5}-\frac{1}{2}a^{4}+\frac{1}{10}a^{3}-\frac{9}{20}a^{2}+\frac{9}{80}a+\frac{7}{40}$, $\frac{1}{5120}a^{26}-\frac{1}{2560}a^{25}-\frac{1}{5120}a^{24}+\frac{1}{2560}a^{23}-\frac{1}{640}a^{22}+\frac{3}{640}a^{21}+\frac{11}{5120}a^{20}+\frac{9}{640}a^{19}-\frac{61}{5120}a^{18}-\frac{23}{320}a^{17}-\frac{239}{5120}a^{16}+\frac{67}{320}a^{15}-\frac{271}{640}a^{14}+\frac{321}{2560}a^{13}+\frac{1729}{5120}a^{12}+\frac{1229}{2560}a^{11}-\frac{2083}{5120}a^{10}+\frac{19}{80}a^{9}-\frac{863}{2560}a^{8}-\frac{1}{5}a^{7}-\frac{257}{640}a^{6}-\frac{17}{80}a^{5}-\frac{11}{80}a^{4}-\frac{21}{160}a^{3}-\frac{11}{320}a^{2}-\frac{11}{40}a+\frac{9}{32}$, $\frac{1}{25600}a^{27}+\frac{3}{25600}a^{25}-\frac{1}{6400}a^{24}-\frac{1}{1280}a^{23}-\frac{3}{6400}a^{22}+\frac{147}{25600}a^{21}+\frac{67}{12800}a^{20}+\frac{283}{25600}a^{19}-\frac{59}{12800}a^{18}-\frac{10503}{25600}a^{17}-\frac{181}{2560}a^{16}+\frac{93}{400}a^{15}-\frac{3581}{12800}a^{14}-\frac{12683}{25600}a^{13}+\frac{413}{1280}a^{12}+\frac{157}{5120}a^{11}+\frac{4451}{12800}a^{10}+\frac{993}{12800}a^{9}-\frac{239}{640}a^{8}+\frac{309}{3200}a^{7}+\frac{77}{800}a^{6}+\frac{77}{800}a^{5}+\frac{171}{800}a^{4}-\frac{363}{1600}a^{3}+\frac{53}{800}a^{2}-\frac{199}{800}a-\frac{21}{100}$, $\frac{1}{102400}a^{28}-\frac{1}{51200}a^{27}-\frac{7}{102400}a^{26}+\frac{3}{10240}a^{25}-\frac{1}{51200}a^{24}+\frac{7}{25600}a^{23}-\frac{69}{102400}a^{22}-\frac{1}{2560}a^{21}+\frac{77}{20480}a^{20}+\frac{21}{6400}a^{19}+\frac{2143}{102400}a^{18}-\frac{987}{3200}a^{17}-\frac{22239}{51200}a^{16}+\frac{24617}{51200}a^{15}-\frac{34319}{102400}a^{14}+\frac{11223}{51200}a^{13}-\frac{4597}{20480}a^{12}-\frac{7347}{25600}a^{11}-\frac{4027}{25600}a^{10}+\frac{63}{6400}a^{9}-\frac{5277}{25600}a^{8}+\frac{69}{160}a^{7}-\frac{1249}{6400}a^{6}+\frac{527}{3200}a^{5}-\frac{3187}{6400}a^{4}-\frac{567}{1600}a^{3}+\frac{51}{320}a^{2}-\frac{9}{40}a-\frac{447}{1600}$, $\frac{1}{1331200}a^{29}+\frac{1}{665600}a^{28}+\frac{21}{1331200}a^{27}+\frac{21}{665600}a^{26}+\frac{193}{665600}a^{25}+\frac{3}{25600}a^{24}-\frac{1317}{1331200}a^{23}-\frac{267}{332800}a^{22}+\frac{4717}{1331200}a^{21}-\frac{139}{66560}a^{20}-\frac{2113}{266240}a^{19}+\frac{1419}{66560}a^{18}-\frac{61861}{665600}a^{17}+\frac{322541}{665600}a^{16}+\frac{419849}{1331200}a^{15}+\frac{56469}{665600}a^{14}-\frac{179629}{1331200}a^{13}+\frac{77829}{166400}a^{12}-\frac{13051}{33280}a^{11}-\frac{13701}{41600}a^{10}-\frac{13723}{66560}a^{9}-\frac{4739}{10400}a^{8}+\frac{25873}{83200}a^{7}+\frac{2161}{41600}a^{6}-\frac{9787}{83200}a^{5}+\frac{319}{800}a^{4}+\frac{203}{520}a^{3}-\frac{2589}{5200}a^{2}+\frac{9351}{20800}a-\frac{609}{2600}$, $\frac{1}{69222400}a^{30}-\frac{7}{34611200}a^{29}-\frac{297}{69222400}a^{28}+\frac{111}{6922240}a^{27}+\frac{63}{1331200}a^{26}-\frac{2553}{8652800}a^{25}+\frac{191}{69222400}a^{24}+\frac{1439}{17305600}a^{23}+\frac{8807}{13844480}a^{22}-\frac{24087}{17305600}a^{21}+\frac{277093}{69222400}a^{20}+\frac{8287}{692224}a^{19}-\frac{75089}{17305600}a^{18}+\frac{3062741}{34611200}a^{17}+\frac{91105}{212992}a^{16}+\frac{7700487}{34611200}a^{15}-\frac{6493867}{69222400}a^{14}-\frac{7872759}{17305600}a^{13}-\frac{14978361}{34611200}a^{12}+\frac{3968837}{8652800}a^{11}+\frac{306801}{17305600}a^{10}-\frac{85361}{332800}a^{9}-\frac{2357689}{8652800}a^{8}-\frac{604391}{2163200}a^{7}-\frac{810657}{4326400}a^{6}-\frac{92737}{1081600}a^{5}+\frac{40037}{86528}a^{4}+\frac{120611}{540800}a^{3}+\frac{11763}{43264}a^{2}-\frac{4133}{270400}a-\frac{192957}{540800}$, $\frac{1}{63\!\cdots\!00}a^{31}+\frac{18\!\cdots\!43}{70\!\cdots\!00}a^{30}+\frac{20\!\cdots\!97}{63\!\cdots\!00}a^{29}-\frac{22\!\cdots\!29}{63\!\cdots\!00}a^{28}-\frac{45\!\cdots\!03}{29\!\cdots\!00}a^{27}-\frac{30\!\cdots\!67}{39\!\cdots\!40}a^{26}-\frac{17\!\cdots\!51}{58\!\cdots\!00}a^{25}+\frac{17\!\cdots\!61}{63\!\cdots\!00}a^{24}+\frac{85\!\cdots\!51}{63\!\cdots\!00}a^{23}+\frac{36\!\cdots\!17}{49\!\cdots\!00}a^{22}-\frac{27\!\cdots\!03}{63\!\cdots\!00}a^{21}+\frac{37\!\cdots\!83}{63\!\cdots\!00}a^{20}+\frac{37\!\cdots\!39}{39\!\cdots\!00}a^{19}-\frac{78\!\cdots\!43}{31\!\cdots\!00}a^{18}-\frac{87\!\cdots\!13}{63\!\cdots\!00}a^{17}-\frac{29\!\cdots\!83}{12\!\cdots\!80}a^{16}+\frac{75\!\cdots\!09}{20\!\cdots\!00}a^{15}+\frac{23\!\cdots\!91}{63\!\cdots\!00}a^{14}+\frac{26\!\cdots\!83}{59\!\cdots\!20}a^{13}+\frac{89\!\cdots\!13}{29\!\cdots\!00}a^{12}+\frac{48\!\cdots\!47}{15\!\cdots\!00}a^{11}-\frac{24\!\cdots\!29}{15\!\cdots\!00}a^{10}-\frac{63\!\cdots\!39}{79\!\cdots\!00}a^{9}-\frac{35\!\cdots\!41}{79\!\cdots\!00}a^{8}-\frac{43\!\cdots\!39}{39\!\cdots\!00}a^{7}-\frac{39\!\cdots\!37}{55\!\cdots\!60}a^{6}+\frac{39\!\cdots\!03}{19\!\cdots\!00}a^{5}-\frac{23\!\cdots\!43}{19\!\cdots\!00}a^{4}+\frac{27\!\cdots\!41}{99\!\cdots\!00}a^{3}+\frac{22\!\cdots\!81}{76\!\cdots\!00}a^{2}+\frac{12\!\cdots\!29}{49\!\cdots\!00}a+\frac{24\!\cdots\!87}{49\!\cdots\!00}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

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

Class group and class number

not computed

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Relative class number: data not computed

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{30265437914002297976796113391006075585639269}{426728225238093518912661691353729263799801036800} a^{31} - \frac{1658598239559302828514879639985058417033}{11812873027297462044974578987756872544563200} a^{30} - \frac{1060062770571493317971592431880468568068353629}{426728225238093518912661691353729263799801036800} a^{29} + \frac{16117948145837184896594377952249832542382275}{4267282252380935189126616913537292637998010368} a^{28} + \frac{400719657849084067473143532963884017631396039}{9698368755411216338924129348948392359086387200} a^{27} - \frac{2690784944323398808611482264575718722383121541}{53341028154761689864082711419216157974975129600} a^{26} - \frac{15977000278323337722247205575495010354826939943}{38793475021644865355696517395793569436345548800} a^{25} + \frac{3949882120165515163709570463546937043684713497}{8534564504761870378253233827074585275996020736} a^{24} + \frac{1163740838104883646912165288290351135198326199643}{426728225238093518912661691353729263799801036800} a^{23} - \frac{148558244080153495501934613113811863988277760291}{42672822523809351891266169135372926379980103680} a^{22} - \frac{5398036445099750937206610519474430019903956070499}{426728225238093518912661691353729263799801036800} a^{21} + \frac{4484388619838219857959978502481505083487267343887}{213364112619046759456330845676864631899900518400} a^{20} + \frac{2245102806940282173418157439040722746647384434861}{53341028154761689864082711419216157974975129600} a^{19} - \frac{20406812755967832248272016000696070877795270354603}{213364112619046759456330845676864631899900518400} a^{18} - \frac{36433765585461546158507987031925266331533196280843}{426728225238093518912661691353729263799801036800} a^{17} + \frac{535772025667508138467653997708571684286886009427}{1666907129836302808252584731850504936717972800} a^{16} + \frac{223391975143150985542357347855610499876115014031}{13765426620583661900408441656571911735477452800} a^{15} - \frac{7179340905270747418439740231775136023133553383001}{8534564504761870378253233827074585275996020736} a^{14} + \frac{1049446062008473460091855886925642852190531963563}{1994057127280810836040475193241725531774771200} a^{13} + \frac{11360655294320030421564509817619913808400617846859}{9698368755411216338924129348948392359086387200} a^{12} - \frac{172767533940109847221386106647840733490762524811891}{106682056309523379728165422838432315949950259200} a^{11} - \frac{9686665090937865823492754438434425920192900141783}{53341028154761689864082711419216157974975129600} a^{10} + \frac{256253579576724460004193905919727320956135515838649}{53341028154761689864082711419216157974975129600} a^{9} - \frac{5888921984728749728346695723481203235505194944873}{1066820563095233797281654228384323159499502592} a^{8} + \frac{43421609684775907683093743461712319054694723287127}{5334102815476168986408271141921615797497512960} a^{7} - \frac{986130596087330940008233336891461206544946334577}{242459218885280408473103233723709808977159680} a^{6} + \frac{55891760599091639353632536960850004569645545007171}{13335257038690422466020677854804039493743782400} a^{5} - \frac{6529774794850859106827205953676620188207515834377}{6667628519345211233010338927402019746871891200} a^{4} + \frac{5616701541116906509296249679837750299886683329311}{6667628519345211233010338927402019746871891200} a^{3} - \frac{73050097957605966022666043823156766048683156533}{3333814259672605616505169463701009873435945600} a^{2} + \frac{24202853120131914227780549624730160555291612357}{666762851934521123301033892740201974687189120} a + \frac{824658653942296774363708821379465805219787929}{1666907129836302808252584731850504936717972800} \)  (order $6$) Copy content Toggle raw display
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot R \cdot h}{6\cdot\sqrt{68145128239473945155058695587402151362560000000000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 35*x^30 + 54*x^29 + 582*x^28 - 724*x^27 - 5801*x^26 + 6656*x^25 + 38405*x^24 - 49984*x^23 - 177937*x^22 + 300832*x^21 + 590114*x^20 - 1365710*x^19 - 1185291*x^18 + 4579318*x^17 + 154147*x^16 - 11942188*x^15 + 7662420*x^14 + 16564120*x^13 - 23303048*x^12 - 2384240*x^11 + 68187376*x^10 - 79122656*x^9 + 115265728*x^8 - 58417728*x^7 + 58682560*x^6 - 14078848*x^5 + 11490944*x^4 - 313088*x^3 + 444416*x^2 + 256)
 
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^32 - 2*x^31 - 35*x^30 + 54*x^29 + 582*x^28 - 724*x^27 - 5801*x^26 + 6656*x^25 + 38405*x^24 - 49984*x^23 - 177937*x^22 + 300832*x^21 + 590114*x^20 - 1365710*x^19 - 1185291*x^18 + 4579318*x^17 + 154147*x^16 - 11942188*x^15 + 7662420*x^14 + 16564120*x^13 - 23303048*x^12 - 2384240*x^11 + 68187376*x^10 - 79122656*x^9 + 115265728*x^8 - 58417728*x^7 + 58682560*x^6 - 14078848*x^5 + 11490944*x^4 - 313088*x^3 + 444416*x^2 + 256, 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^32 - 2*x^31 - 35*x^30 + 54*x^29 + 582*x^28 - 724*x^27 - 5801*x^26 + 6656*x^25 + 38405*x^24 - 49984*x^23 - 177937*x^22 + 300832*x^21 + 590114*x^20 - 1365710*x^19 - 1185291*x^18 + 4579318*x^17 + 154147*x^16 - 11942188*x^15 + 7662420*x^14 + 16564120*x^13 - 23303048*x^12 - 2384240*x^11 + 68187376*x^10 - 79122656*x^9 + 115265728*x^8 - 58417728*x^7 + 58682560*x^6 - 14078848*x^5 + 11490944*x^4 - 313088*x^3 + 444416*x^2 + 256);
 
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^32 - 2*x^31 - 35*x^30 + 54*x^29 + 582*x^28 - 724*x^27 - 5801*x^26 + 6656*x^25 + 38405*x^24 - 49984*x^23 - 177937*x^22 + 300832*x^21 + 590114*x^20 - 1365710*x^19 - 1185291*x^18 + 4579318*x^17 + 154147*x^16 - 11942188*x^15 + 7662420*x^14 + 16564120*x^13 - 23303048*x^12 - 2384240*x^11 + 68187376*x^10 - 79122656*x^9 + 115265728*x^8 - 58417728*x^7 + 58682560*x^6 - 14078848*x^5 + 11490944*x^4 - 313088*x^3 + 444416*x^2 + 256);
 
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

$D_4^2:C_2^3$ (as 32T12882):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 512
The 80 conjugacy class representatives for $D_4^2:C_2^3$
Character table for $D_4^2:C_2^3$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{10}) \), 4.0.417600.1, 4.4.46400.1, 4.4.725.1, 4.0.6525.1, \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.22181900625.1, 8.0.273850625.1, 8.0.1121692160000.1, 8.8.90857064960000.1, 8.0.207360000.1, 8.0.174389760000.40, 8.0.42575625.1, 8.0.174389760000.27, 8.0.174389760000.12, 8.0.174389760000.25, 8.8.2152960000.1, 16.0.30411788392857600000000.2, 16.0.492036715337375390625.1, 16.0.8255006253145659801600000000.5, 16.0.8255006253145659801600000000.1, 16.0.8255006253145659801600000000.2, 16.16.8255006253145659801600000000.1, 16.0.1258193301805465600000000.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 32 siblings: 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 R ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.2.0.1}{2} }^{16}$ ${\href{/padicField/17.8.0.1}{8} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ R ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{8}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.8.12.1$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
\(3\) Copy content Toggle raw display 3.16.8.1$x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$
3.16.8.1$x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(29\) Copy content Toggle raw display 29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(521\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$