Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 9*x^30 + 40*x^28 + 119*x^26 + 271*x^24 + 495*x^22 + 752*x^20 + 1105*x^18 + 1941*x^16 + 4420*x^14 + 12032*x^12 + 31680*x^10 + 69376*x^8 + 121856*x^6 + 163840*x^4 + 147456*x^2 + 65536)
gp: K = bnfinit(y^32 + 9*y^30 + 40*y^28 + 119*y^26 + 271*y^24 + 495*y^22 + 752*y^20 + 1105*y^18 + 1941*y^16 + 4420*y^14 + 12032*y^12 + 31680*y^10 + 69376*y^8 + 121856*y^6 + 163840*y^4 + 147456*y^2 + 65536, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 9*x^30 + 40*x^28 + 119*x^26 + 271*x^24 + 495*x^22 + 752*x^20 + 1105*x^18 + 1941*x^16 + 4420*x^14 + 12032*x^12 + 31680*x^10 + 69376*x^8 + 121856*x^6 + 163840*x^4 + 147456*x^2 + 65536);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 + 9*x^30 + 40*x^28 + 119*x^26 + 271*x^24 + 495*x^22 + 752*x^20 + 1105*x^18 + 1941*x^16 + 4420*x^14 + 12032*x^12 + 31680*x^10 + 69376*x^8 + 121856*x^6 + 163840*x^4 + 147456*x^2 + 65536)
\( x^{32} + 9 x^{30} + 40 x^{28} + 119 x^{26} + 271 x^{24} + 495 x^{22} + 752 x^{20} + 1105 x^{18} + \cdots + 65536 \)
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: |
\(38959704907616347430279100767614402560000000000000000\)
\(\medspace = 2^{32}\cdot 3^{16}\cdot 5^{16}\cdot 29^{8}\cdot 1289^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(44.00\) | 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: | $2\cdot 3^{1/2}5^{1/2}29^{1/2}1289^{1/2}\approx 1497.6181088648734$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(29\), \(1289\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{18}+\frac{1}{4}a^{16}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{19}+\frac{1}{8}a^{17}-\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{8}a^{9}+\frac{1}{8}a^{5}-\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{20}+\frac{1}{16}a^{18}+\frac{7}{16}a^{14}+\frac{7}{16}a^{12}+\frac{7}{16}a^{10}-\frac{1}{2}a^{8}+\frac{1}{16}a^{6}-\frac{3}{16}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{32}a^{21}+\frac{1}{32}a^{19}-\frac{9}{32}a^{15}-\frac{9}{32}a^{13}-\frac{9}{32}a^{11}-\frac{1}{4}a^{9}-\frac{15}{32}a^{7}+\frac{13}{32}a^{5}-\frac{1}{8}a^{3}$, $\frac{1}{64}a^{22}+\frac{1}{64}a^{20}+\frac{23}{64}a^{16}+\frac{23}{64}a^{14}+\frac{23}{64}a^{12}+\frac{3}{8}a^{10}-\frac{15}{64}a^{8}+\frac{13}{64}a^{6}-\frac{1}{16}a^{4}$, $\frac{1}{128}a^{23}+\frac{1}{128}a^{21}+\frac{23}{128}a^{17}+\frac{23}{128}a^{15}-\frac{41}{128}a^{13}+\frac{3}{16}a^{11}-\frac{15}{128}a^{9}-\frac{51}{128}a^{7}+\frac{15}{32}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{256}a^{24}+\frac{1}{256}a^{22}+\frac{23}{256}a^{18}+\frac{23}{256}a^{16}+\frac{87}{256}a^{14}-\frac{13}{32}a^{12}-\frac{15}{256}a^{10}+\frac{77}{256}a^{8}-\frac{17}{64}a^{6}+\frac{1}{4}a^{4}$, $\frac{1}{512}a^{25}+\frac{1}{512}a^{23}+\frac{23}{512}a^{19}+\frac{23}{512}a^{17}+\frac{87}{512}a^{15}-\frac{13}{64}a^{13}+\frac{241}{512}a^{11}-\frac{179}{512}a^{9}+\frac{47}{128}a^{7}-\frac{3}{8}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{1024}a^{26}+\frac{1}{1024}a^{24}+\frac{23}{1024}a^{20}+\frac{23}{1024}a^{18}-\frac{425}{1024}a^{16}-\frac{13}{128}a^{14}-\frac{271}{1024}a^{12}-\frac{179}{1024}a^{10}-\frac{81}{256}a^{8}-\frac{3}{16}a^{6}-\frac{1}{4}a^{4}$, $\frac{1}{2048}a^{27}+\frac{1}{2048}a^{25}+\frac{23}{2048}a^{21}+\frac{23}{2048}a^{19}-\frac{425}{2048}a^{17}-\frac{13}{256}a^{15}-\frac{271}{2048}a^{13}-\frac{179}{2048}a^{11}+\frac{175}{512}a^{9}+\frac{13}{32}a^{7}-\frac{1}{8}a^{5}-\frac{1}{2}a$, $\frac{1}{185495552}a^{28}-\frac{17207}{185495552}a^{26}-\frac{37941}{23186944}a^{24}+\frac{392935}{185495552}a^{22}+\frac{2685519}{185495552}a^{20}-\frac{7040193}{185495552}a^{18}-\frac{1013983}{2898368}a^{16}-\frac{49397919}{185495552}a^{14}+\frac{63724245}{185495552}a^{12}-\frac{14076515}{46373888}a^{10}-\frac{4791365}{11593472}a^{8}-\frac{52611}{362296}a^{6}-\frac{18325}{362296}a^{4}+\frac{67033}{181148}a^{2}+\frac{4}{45287}$, $\frac{1}{370991104}a^{29}-\frac{17207}{370991104}a^{27}-\frac{37941}{46373888}a^{25}+\frac{392935}{370991104}a^{23}+\frac{2685519}{370991104}a^{21}-\frac{7040193}{370991104}a^{19}-\frac{1013983}{5796736}a^{17}-\frac{49397919}{370991104}a^{15}-\frac{121771307}{370991104}a^{13}+\frac{32297373}{92747776}a^{11}-\frac{4791365}{23186944}a^{9}-\frac{52611}{724592}a^{7}-\frac{18325}{724592}a^{5}+\frac{67033}{362296}a^{3}+\frac{2}{45287}a$, $\frac{1}{156558245888}a^{30}-\frac{411}{156558245888}a^{28}+\frac{1218013}{39139561472}a^{26}+\frac{262868263}{156558245888}a^{24}-\frac{573937389}{156558245888}a^{22}-\frac{1464794173}{156558245888}a^{20}-\frac{196890713}{1701720064}a^{18}-\frac{45839444799}{156558245888}a^{16}+\frac{53659721457}{156558245888}a^{14}+\frac{1802344455}{4892445184}a^{12}-\frac{324339627}{1223111296}a^{10}+\frac{436497351}{2446222592}a^{8}-\frac{57366079}{305777824}a^{6}-\frac{40518243}{152888912}a^{4}-\frac{698732}{9555557}a^{2}+\frac{1194258}{9555557}$, $\frac{1}{313116491776}a^{31}-\frac{411}{313116491776}a^{29}+\frac{1218013}{78279122944}a^{27}+\frac{262868263}{313116491776}a^{25}-\frac{573937389}{313116491776}a^{23}-\frac{1464794173}{313116491776}a^{21}-\frac{196890713}{3403440128}a^{19}-\frac{45839444799}{313116491776}a^{17}-\frac{102898524431}{313116491776}a^{15}-\frac{3090100729}{9784890368}a^{13}-\frac{324339627}{2446222592}a^{11}-\frac{2009725241}{4892445184}a^{9}+\frac{248411745}{611555648}a^{7}+\frac{112370669}{305777824}a^{5}-\frac{349366}{9555557}a^{3}-\frac{8361299}{19111114}a$
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
$C_{38}$, which has order $38$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $38$ (assuming GRH)
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: |
\( \frac{52117223}{313116491776} a^{31} + \frac{452695779}{313116491776} a^{29} + \frac{458621979}{78279122944} a^{27} + \frac{4884892401}{313116491776} a^{25} + \frac{10385453349}{313116491776} a^{23} + \frac{17393024821}{313116491776} a^{21} + \frac{6115947847}{78279122944} a^{19} + \frac{36342202471}{313116491776} a^{17} + \frac{71922073207}{313116491776} a^{15} + \frac{5506386647}{9784890368} a^{13} + \frac{7867295333}{4892445184} a^{11} + \frac{2535521851}{611555648} a^{9} + \frac{116084461}{13898992} a^{7} + \frac{4105745177}{305777824} a^{5} + \frac{13475301}{868687} a^{3} + \frac{95650719}{9555557} a \)
(order $12$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{485373}{2446222592}a^{30}+\frac{2570531}{4892445184}a^{28}-\frac{3420985}{9784890368}a^{26}-\frac{173101}{54664192}a^{24}-\frac{54198135}{4892445184}a^{22}-\frac{273256979}{9784890368}a^{20}-\frac{369343655}{9784890368}a^{18}-\frac{341282273}{9784890368}a^{16}-\frac{313481533}{4892445184}a^{14}-\frac{23212227}{889535488}a^{12}-\frac{813291119}{9784890368}a^{10}-\frac{2400453371}{2446222592}a^{8}-\frac{459501299}{152888912}a^{6}-\frac{64698213}{9555557}a^{4}-\frac{465636093}{38222228}a^{2}-\frac{104898921}{9555557}$, $\frac{257293}{370991104}a^{30}+\frac{1693785}{370991104}a^{28}+\frac{1403935}{92747776}a^{26}+\frac{13504411}{370991104}a^{24}+\frac{26512831}{370991104}a^{22}+\frac{39939599}{370991104}a^{20}+\frac{1214549}{8431616}a^{18}+\frac{94455789}{370991104}a^{16}+\frac{192341301}{370991104}a^{14}+\frac{64742915}{46373888}a^{12}+\frac{23700439}{5796736}a^{10}+\frac{6901933}{724592}a^{8}+\frac{50699069}{2898368}a^{6}+\frac{9266459}{362296}a^{4}+\frac{1062948}{45287}a^{2}+\frac{365690}{45287}$, $\frac{52117223}{313116491776}a^{31}+\frac{452695779}{313116491776}a^{29}+\frac{458621979}{78279122944}a^{27}+\frac{4884892401}{313116491776}a^{25}+\frac{10385453349}{313116491776}a^{23}+\frac{17393024821}{313116491776}a^{21}+\frac{6115947847}{78279122944}a^{19}+\frac{36342202471}{313116491776}a^{17}+\frac{71922073207}{313116491776}a^{15}+\frac{5506386647}{9784890368}a^{13}+\frac{7867295333}{4892445184}a^{11}+\frac{2535521851}{611555648}a^{9}+\frac{116084461}{13898992}a^{7}+\frac{4105745177}{305777824}a^{5}+\frac{13475301}{868687}a^{3}+\frac{95650719}{9555557}a+1$, $\frac{1075157}{741982208}a^{30}+\frac{8114457}{741982208}a^{28}+\frac{7459957}{185495552}a^{26}+\frac{76548851}{741982208}a^{24}+\frac{156059263}{741982208}a^{22}+\frac{250479727}{741982208}a^{20}+\frac{7802323}{16863232}a^{18}+\frac{550196117}{741982208}a^{16}+\frac{1102043477}{741982208}a^{14}+\frac{174924327}{46373888}a^{12}+\frac{127035983}{11593472}a^{10}+\frac{39146601}{1449184}a^{8}+\frac{76081683}{1449184}a^{6}+\frac{58733595}{724592}a^{4}+\frac{3905370}{45287}a^{2}+\frac{2034841}{45287}$, $\frac{79647507}{313116491776}a^{31}+\frac{427707487}{313116491776}a^{29}+\frac{1687237}{437313536}a^{27}+\frac{2663654245}{313116491776}a^{25}+\frac{4915310985}{313116491776}a^{23}+\frac{6575790105}{313116491776}a^{21}+\frac{2144024123}{78279122944}a^{19}+\frac{17877918163}{313116491776}a^{17}+\frac{36293476803}{313116491776}a^{15}+\frac{862412769}{2446222592}a^{13}+\frac{2557656597}{2446222592}a^{11}+\frac{10937792585}{4892445184}a^{9}+\frac{2310110339}{611555648}a^{7}+\frac{757892617}{152888912}a^{5}+\frac{220039999}{76444456}a^{3}-\frac{12679061}{9555557}a$, $\frac{28036407}{156558245888}a^{31}+\frac{209152941}{156558245888}a^{30}+\frac{104169357}{156558245888}a^{29}+\frac{1608630625}{156558245888}a^{28}+\frac{99357151}{78279122944}a^{27}+\frac{1534443921}{39139561472}a^{26}+\frac{235922041}{156558245888}a^{25}+\frac{89501945}{874627072}a^{24}+\frac{67449291}{156558245888}a^{23}+\frac{32973332407}{156558245888}a^{22}-\frac{758816061}{156558245888}a^{21}+\frac{54008903911}{156558245888}a^{20}-\frac{549971315}{78279122944}a^{19}+\frac{18731154549}{39139561472}a^{18}+\frac{469977359}{156558245888}a^{17}+\frac{116331051949}{156558245888}a^{16}+\frac{1530596385}{156558245888}a^{15}+\frac{232454921725}{156558245888}a^{14}+\frac{8237765781}{78279122944}a^{13}+\frac{824393081}{222383872}a^{12}+\frac{3033090037}{9784890368}a^{11}+\frac{26113314187}{2446222592}a^{10}+\frac{446732389}{1223111296}a^{9}+\frac{65547939111}{2446222592}a^{8}-\frac{560043}{4834432}a^{7}+\frac{32388604315}{611555648}a^{6}-\frac{382329631}{305777824}a^{5}+\frac{12676031871}{152888912}a^{4}-\frac{7680815}{1737374}a^{3}+\frac{3511031845}{38222228}a^{2}-\frac{46801685}{9555557}a+\frac{488397832}{9555557}$, $\frac{50790841}{313116491776}a^{31}-\frac{18543319}{156558245888}a^{30}+\frac{298115517}{313116491776}a^{29}-\frac{209304259}{156558245888}a^{28}+\frac{213554205}{78279122944}a^{27}-\frac{227958163}{39139561472}a^{26}+\frac{1840841935}{313116491776}a^{25}-\frac{2484623729}{156558245888}a^{24}+\frac{3564207547}{313116491776}a^{23}-\frac{5156047173}{156558245888}a^{22}+\frac{4849721163}{313116491776}a^{21}-\frac{9012721125}{156558245888}a^{20}+\frac{128231363}{7116283904}a^{19}-\frac{3048849967}{39139561472}a^{18}+\frac{13081247193}{313116491776}a^{17}-\frac{17620883815}{156558245888}a^{16}+\frac{28158471721}{313116491776}a^{15}-\frac{35964089879}{156558245888}a^{14}+\frac{603009707}{2446222592}a^{13}-\frac{1307631681}{2446222592}a^{12}+\frac{7211052341}{9784890368}a^{11}-\frac{15369826315}{9784890368}a^{10}+\frac{1909973277}{1223111296}a^{9}-\frac{10244816341}{2446222592}a^{8}+\frac{1558109801}{611555648}a^{7}-\frac{227255671}{26589376}a^{6}+\frac{266311229}{76444456}a^{5}-\frac{1041323663}{76444456}a^{4}+\frac{45515575}{19111114}a^{3}-\frac{625194767}{38222228}a^{2}-\frac{5026207}{9555557}a-\frac{87877479}{9555557}$, $\frac{22851}{67452928}a^{31}+\frac{50961}{67452928}a^{30}+\frac{132669}{67452928}a^{29}+\frac{429717}{67452928}a^{28}+\frac{200689}{33726464}a^{27}+\frac{422917}{16863232}a^{26}+\frac{896653}{67452928}a^{25}+\frac{4503639}{67452928}a^{24}+\frac{1629083}{67452928}a^{23}+\frac{9366691}{67452928}a^{22}+\frac{2294419}{67452928}a^{21}+\frac{15509139}{67452928}a^{20}+\frac{1497723}{33726464}a^{19}+\frac{5373225}{16863232}a^{18}+\frac{5732411}{67452928}a^{17}+\frac{32844049}{67452928}a^{16}+\frac{12366369}{67452928}a^{15}+\frac{65214625}{67452928}a^{14}+\frac{17777959}{33726464}a^{13}+\frac{2504123}{1053952}a^{12}+\frac{1628855}{1053952}a^{11}+\frac{7239555}{1053952}a^{10}+\frac{1813375}{526976}a^{9}+\frac{2303885}{131744}a^{8}+\frac{1563485}{263488}a^{7}+\frac{9224027}{263488}a^{6}+\frac{1034015}{131744}a^{5}+\frac{3654607}{65872}a^{4}+\frac{183137}{32936}a^{3}+\frac{258402}{4117}a^{2}-\frac{120}{4117}a+\frac{147624}{4117}$, $\frac{9437213}{78279122944}a^{31}+\frac{65003657}{39139561472}a^{29}+\frac{588427713}{78279122944}a^{27}+\frac{1671218135}{78279122944}a^{25}+\frac{1809419303}{39139561472}a^{23}+\frac{3176190393}{39139561472}a^{21}+\frac{8626508255}{78279122944}a^{19}+\frac{1135377499}{7116283904}a^{17}+\frac{1130164949}{3558141952}a^{15}+\frac{2421465503}{3403440128}a^{13}+\frac{20692571807}{9784890368}a^{11}+\frac{869951319}{152888912}a^{9}+\frac{7263058589}{611555648}a^{7}+\frac{5984271759}{305777824}a^{5}+\frac{233463607}{9555557}a^{3}+\frac{140018859}{9555557}a$, $\frac{46406907}{313116491776}a^{31}-\frac{69820759}{78279122944}a^{30}+\frac{232227323}{313116491776}a^{29}-\frac{398517131}{78279122944}a^{28}+\frac{43029583}{19569780736}a^{27}-\frac{289388315}{19569780736}a^{26}+\frac{1592759101}{313116491776}a^{25}-\frac{2601550089}{78279122944}a^{24}+\frac{2929892941}{313116491776}a^{23}-\frac{4727037181}{78279122944}a^{22}+\frac{3902205149}{313116491776}a^{21}-\frac{6241199557}{78279122944}a^{20}+\frac{694411163}{39139561472}a^{19}-\frac{2080280919}{19569780736}a^{18}+\frac{10419850779}{313116491776}a^{17}-\frac{17199913295}{78279122944}a^{16}+\frac{20184564031}{313116491776}a^{15}-\frac{35568309983}{78279122944}a^{14}+\frac{16156595857}{78279122944}a^{13}-\frac{1675677571}{1223111296}a^{12}+\frac{11742202427}{19569780736}a^{11}-\frac{39193049913}{9784890368}a^{10}+\frac{1532757643}{1223111296}a^{9}-\frac{20900472437}{2446222592}a^{8}+\frac{2694611441}{1223111296}a^{7}-\frac{385195581}{26589376}a^{6}+\frac{943572897}{305777824}a^{5}-\frac{1437637145}{76444456}a^{4}+\frac{63657555}{38222228}a^{3}-\frac{431492019}{38222228}a^{2}+\frac{315447}{1737374}a+\frac{27738331}{9555557}$, $\frac{198352525}{313116491776}a^{31}+\frac{422941575}{156558245888}a^{30}+\frac{1785307917}{313116491776}a^{29}+\frac{2811563363}{156558245888}a^{28}+\frac{19347589}{850860032}a^{27}+\frac{2433105003}{39139561472}a^{26}+\frac{19333725019}{313116491776}a^{25}+\frac{24193082993}{156558245888}a^{24}+\frac{40670133035}{313116491776}a^{23}+\frac{47804043877}{156558245888}a^{22}+\frac{68113657915}{313116491776}a^{21}+\frac{74693052501}{156558245888}a^{20}+\frac{11729675841}{39139561472}a^{19}+\frac{25683550279}{39139561472}a^{18}+\frac{142537732493}{313116491776}a^{17}+\frac{170386272231}{156558245888}a^{16}+\frac{281159871337}{313116491776}a^{15}+\frac{344932988343}{156558245888}a^{14}+\frac{15383036653}{7116283904}a^{13}+\frac{28597021777}{4892445184}a^{12}+\frac{123025970963}{19569780736}a^{11}+\frac{82400421051}{4892445184}a^{10}+\frac{78767862597}{4892445184}a^{9}+\frac{98399570149}{2446222592}a^{8}+\frac{40200714643}{1223111296}a^{7}+\frac{46552379035}{611555648}a^{6}+\frac{8041478955}{152888912}a^{5}+\frac{17326776959}{152888912}a^{4}+\frac{4635448677}{76444456}a^{3}+\frac{2170060549}{19111114}a^{2}+\frac{678398819}{19111114}a+\frac{489647891}{9555557}$, $\frac{181727}{212715008}a^{31}-\frac{269554755}{78279122944}a^{30}+\frac{591524859}{78279122944}a^{29}-\frac{2019134023}{78279122944}a^{28}+\frac{2481775339}{78279122944}a^{27}-\frac{1890862395}{19569780736}a^{26}+\frac{209666621}{2446222592}a^{25}-\frac{19505053741}{78279122944}a^{24}+\frac{14252755837}{78279122944}a^{23}-\frac{39974121377}{78279122944}a^{22}+\frac{24231730893}{78279122944}a^{21}-\frac{64559333369}{78279122944}a^{20}+\frac{33777356973}{78279122944}a^{19}-\frac{22212387567}{19569780736}a^{18}+\frac{6274288997}{9784890368}a^{17}-\frac{140137629451}{78279122944}a^{16}+\frac{100120579483}{78279122944}a^{15}-\frac{280588741995}{78279122944}a^{14}+\frac{21434933741}{7116283904}a^{13}-\frac{44339672301}{4892445184}a^{12}+\frac{169287213969}{19569780736}a^{11}-\frac{11191931489}{425430016}a^{10}+\frac{110548858239}{4892445184}a^{9}-\frac{160101415733}{2446222592}a^{8}+\frac{1751661581}{38222228}a^{7}-\frac{3559169273}{27797984}a^{6}+\frac{22694172003}{305777824}a^{5}-\frac{30512908461}{152888912}a^{4}+\frac{3347004717}{38222228}a^{3}-\frac{186683160}{868687}a^{2}+\frac{510552971}{9555557}a-\frac{1110778958}{9555557}$, $\frac{3858817}{1483964416}a^{31}-\frac{2130369}{741982208}a^{30}+\frac{23379489}{1483964416}a^{29}-\frac{15097381}{741982208}a^{28}+\frac{5048173}{92747776}a^{27}-\frac{13510125}{185495552}a^{26}+\frac{201460551}{1483964416}a^{25}-\frac{136606119}{741982208}a^{24}+\frac{399516023}{1483964416}a^{23}-\frac{275221843}{741982208}a^{22}+\frac{619615111}{1483964416}a^{21}-\frac{434976195}{741982208}a^{20}+\frac{111007417}{185495552}a^{19}-\frac{13591795}{16863232}a^{18}+\frac{1434805569}{1483964416}a^{17}-\frac{979928129}{741982208}a^{16}+\frac{2896691565}{1483964416}a^{15}-\frac{1950653361}{741982208}a^{14}+\frac{1952120195}{370991104}a^{13}-\frac{158692561}{23186944}a^{12}+\frac{1368580475}{92747776}a^{11}-\frac{229481687}{11593472}a^{10}+\frac{816803795}{23186944}a^{9}-\frac{557582635}{11593472}a^{8}+\frac{2213203}{32936}a^{7}-\frac{133796671}{1449184}a^{6}+\frac{145114627}{1449184}a^{5}-\frac{51090031}{362296}a^{4}+\frac{1646037}{16468}a^{3}-\frac{13038547}{90574}a^{2}+\frac{2472287}{45287}a-\frac{3291716}{45287}$, $\frac{84101043}{156558245888}a^{31}-\frac{14123503}{39139561472}a^{30}+\frac{390181741}{156558245888}a^{29}-\frac{24093}{13666048}a^{28}+\frac{438431289}{78279122944}a^{27}-\frac{177273561}{39139561472}a^{26}+\frac{1585370349}{156558245888}a^{25}-\frac{397462401}{39139561472}a^{24}+\frac{2046761323}{156558245888}a^{23}-\frac{951013}{55595968}a^{22}+\frac{953234835}{156558245888}a^{21}-\frac{600653}{27332096}a^{20}+\frac{2913217}{437313536}a^{19}-\frac{1273849463}{39139561472}a^{18}+\frac{7844409563}{156558245888}a^{17}-\frac{2706833999}{39139561472}a^{16}+\frac{18670933393}{156558245888}a^{15}-\frac{1407660953}{9784890368}a^{14}+\frac{3565451533}{7116283904}a^{13}-\frac{17470823849}{39139561472}a^{12}+\frac{14308998701}{9784890368}a^{11}-\frac{1133628379}{889535488}a^{10}+\frac{6017233379}{2446222592}a^{9}-\frac{6077373517}{2446222592}a^{8}+\frac{1790843489}{611555648}a^{7}-\frac{2556668117}{611555648}a^{6}+\frac{329605999}{305777824}a^{5}-\frac{356163521}{76444456}a^{4}-\frac{22009183}{3323672}a^{3}-\frac{36111287}{19111114}a^{2}-\frac{105177441}{9555557}a+\frac{10259720}{9555557}$, $\frac{73154655}{78279122944}a^{31}-\frac{47659089}{156558245888}a^{30}+\frac{17734289}{3558141952}a^{29}-\frac{31575279}{6806880256}a^{28}+\frac{1110772283}{78279122944}a^{27}-\frac{418808691}{19569780736}a^{26}+\frac{219138919}{7116283904}a^{25}-\frac{9466532663}{156558245888}a^{24}+\frac{2084270157}{39139561472}a^{23}-\frac{20811252831}{156558245888}a^{22}+\frac{2603922923}{39139561472}a^{21}-\frac{36375781679}{156558245888}a^{20}+\frac{6923506357}{78279122944}a^{19}-\frac{779619241}{2446222592}a^{18}+\frac{15193473619}{78279122944}a^{17}-\frac{72258449297}{156558245888}a^{16}+\frac{16132045517}{39139561472}a^{15}-\frac{140009889285}{156558245888}a^{14}+\frac{101721386323}{78279122944}a^{13}-\frac{78395431485}{39139561472}a^{12}+\frac{36661762631}{9784890368}a^{11}-\frac{57860108957}{9784890368}a^{10}+\frac{38141391187}{4892445184}a^{9}-\frac{39235607615}{2446222592}a^{8}+\frac{15543229825}{1223111296}a^{7}-\frac{1870207469}{55595968}a^{6}+\frac{2338072123}{152888912}a^{5}-\frac{8585805219}{152888912}a^{4}+\frac{476610325}{76444456}a^{3}-\frac{60665933}{868687}a^{2}-\frac{144978777}{19111114}a-\frac{422487849}{9555557}$
| 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: | \( 2224349047480.508 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 2224349047480.508 \cdot 38}{12\cdot\sqrt{38959704907616347430279100767614402560000000000000000}}\cr\approx \mathstrut & 0.210559748495449 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 + 9*x^30 + 40*x^28 + 119*x^26 + 271*x^24 + 495*x^22 + 752*x^20 + 1105*x^18 + 1941*x^16 + 4420*x^14 + 12032*x^12 + 31680*x^10 + 69376*x^8 + 121856*x^6 + 163840*x^4 + 147456*x^2 + 65536)
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 + 9*x^30 + 40*x^28 + 119*x^26 + 271*x^24 + 495*x^22 + 752*x^20 + 1105*x^18 + 1941*x^16 + 4420*x^14 + 12032*x^12 + 31680*x^10 + 69376*x^8 + 121856*x^6 + 163840*x^4 + 147456*x^2 + 65536, 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 + 9*x^30 + 40*x^28 + 119*x^26 + 271*x^24 + 495*x^22 + 752*x^20 + 1105*x^18 + 1941*x^16 + 4420*x^14 + 12032*x^12 + 31680*x^10 + 69376*x^8 + 121856*x^6 + 163840*x^4 + 147456*x^2 + 65536);
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 + 9*x^30 + 40*x^28 + 119*x^26 + 271*x^24 + 495*x^22 + 752*x^20 + 1105*x^18 + 1941*x^16 + 4420*x^14 + 12032*x^12 + 31680*x^10 + 69376*x^8 + 121856*x^6 + 163840*x^4 + 147456*x^2 + 65536);
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
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 | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.8.0.1}{8} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{16}$ | ${\href{/padicField/23.2.0.1}{2} }^{16}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(3\)
| 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 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ | |
|
\(5\)
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(29\)
| 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 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(1289\)
| 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 $4$ | $2$ | $2$ | $2$ |