Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^28 + x^24 - 57*x^20 + 161*x^16 - 912*x^12 + 256*x^8 - 4096*x^4 + 65536)
gp: K = bnfinit(y^32 - y^28 + y^24 - 57*y^20 + 161*y^16 - 912*y^12 + 256*y^8 - 4096*y^4 + 65536, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - x^28 + x^24 - 57*x^20 + 161*x^16 - 912*x^12 + 256*x^8 - 4096*x^4 + 65536);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - x^28 + x^24 - 57*x^20 + 161*x^16 - 912*x^12 + 256*x^8 - 4096*x^4 + 65536)
\( x^{32} - x^{28} + x^{24} - 57x^{20} + 161x^{16} - 912x^{12} + 256x^{8} - 4096x^{4} + 65536 \)
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: |
\(3887187097015424555287785519115204034560000000000000000\)
\(\medspace = 2^{64}\cdot 5^{16}\cdot 29^{8}\cdot 1289^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(50.81\) | 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: | $2^{2}5^{1/2}29^{1/2}1289^{1/2}\approx 1729.300436592786$ | ||
| Ramified primes: |
\(2\), \(5\), \(29\), \(1289\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{14}+\frac{1}{4}a^{10}-\frac{1}{4}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{19}-\frac{1}{8}a^{15}+\frac{1}{8}a^{11}-\frac{1}{8}a^{7}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{20}-\frac{1}{16}a^{16}+\frac{1}{16}a^{12}+\frac{7}{16}a^{8}+\frac{1}{16}a^{4}$, $\frac{1}{32}a^{21}-\frac{1}{32}a^{17}+\frac{1}{32}a^{13}+\frac{7}{32}a^{9}+\frac{1}{32}a^{5}-\frac{1}{2}a$, $\frac{1}{64}a^{22}-\frac{1}{64}a^{18}+\frac{1}{64}a^{14}+\frac{7}{64}a^{10}-\frac{31}{64}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{128}a^{23}-\frac{1}{128}a^{19}+\frac{1}{128}a^{15}-\frac{57}{128}a^{11}+\frac{33}{128}a^{7}-\frac{1}{8}a^{3}$, $\frac{1}{1133312}a^{24}+\frac{26079}{1133312}a^{20}+\frac{365601}{1133312}a^{16}-\frac{309593}{1133312}a^{12}-\frac{441151}{1133312}a^{8}-\frac{12155}{70832}a^{4}+\frac{16}{4427}$, $\frac{1}{2266624}a^{25}+\frac{26079}{2266624}a^{21}+\frac{365601}{2266624}a^{17}+\frac{823719}{2266624}a^{13}+\frac{692161}{2266624}a^{9}-\frac{12155}{141664}a^{5}-\frac{4411}{8854}a$, $\frac{1}{4533248}a^{26}+\frac{26079}{4533248}a^{22}+\frac{365601}{4533248}a^{18}-\frac{1442905}{4533248}a^{14}+\frac{692161}{4533248}a^{10}-\frac{12155}{283328}a^{6}-\frac{4411}{17708}a^{2}$, $\frac{1}{9066496}a^{27}+\frac{26079}{9066496}a^{23}+\frac{365601}{9066496}a^{19}-\frac{1442905}{9066496}a^{15}-\frac{3841087}{9066496}a^{11}+\frac{271173}{566656}a^{7}-\frac{4411}{35416}a^{3}$, $\frac{1}{199462912}a^{28}+\frac{47}{199462912}a^{24}+\frac{315659}{10498048}a^{20}+\frac{59984887}{199462912}a^{16}+\frac{63367665}{199462912}a^{12}-\frac{1239627}{6233216}a^{8}-\frac{92571}{389576}a^{4}+\frac{4957}{48697}$, $\frac{1}{398925824}a^{29}+\frac{47}{398925824}a^{25}+\frac{315659}{20996096}a^{21}+\frac{59984887}{398925824}a^{17}+\frac{63367665}{398925824}a^{13}+\frac{4993589}{12466432}a^{9}-\frac{92571}{779152}a^{5}+\frac{4957}{97394}a$, $\frac{1}{797851648}a^{30}+\frac{47}{797851648}a^{26}+\frac{315659}{41992192}a^{22}+\frac{59984887}{797851648}a^{18}+\frac{63367665}{797851648}a^{14}+\frac{4993589}{24932864}a^{10}-\frac{92571}{1558304}a^{6}-\frac{92437}{194788}a^{2}$, $\frac{1}{1595703296}a^{31}+\frac{47}{1595703296}a^{27}+\frac{315659}{83984384}a^{23}+\frac{59984887}{1595703296}a^{19}-\frac{734483983}{1595703296}a^{15}-\frac{19939275}{49865728}a^{11}+\frac{1465733}{3116608}a^{7}-\frac{92437}{389576}a^{3}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{150}$, which has order $150$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $150$ (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: |
\( \frac{1169}{99731456} a^{29} - \frac{8945}{99731456} a^{25} - \frac{15823}{99731456} a^{21} + \frac{113175}{99731456} a^{17} - \frac{571951}{99731456} a^{13} - \frac{144003}{6233216} a^{9} - \frac{101399}{1558304} a^{5} + \frac{17895}{48697} a \)
(order $8$)
| 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: |
$\frac{9501}{797851648}a^{30}+\frac{127283}{797851648}a^{26}-\frac{122931}{797851648}a^{22}+\frac{971227}{797851648}a^{18}-\frac{2121619}{797851648}a^{14}+\frac{4515}{389576}a^{10}-\frac{214783}{3116608}a^{6}-\frac{5156}{48697}a^{2}$, $\frac{309}{9066496}a^{28}+\frac{107}{9066496}a^{24}+\frac{6933}{9066496}a^{20}-\frac{24237}{9066496}a^{16}-\frac{79563}{9066496}a^{12}-\frac{38123}{566656}a^{8}-\frac{3427}{70832}a^{4}-\frac{2328}{4427}$, $\frac{309}{9066496}a^{28}+\frac{107}{9066496}a^{24}+\frac{6933}{9066496}a^{20}-\frac{24237}{9066496}a^{16}-\frac{79563}{9066496}a^{12}-\frac{38123}{566656}a^{8}-\frac{3427}{70832}a^{4}+\frac{2099}{4427}$, $\frac{30699}{1595703296}a^{31}+\frac{1169}{99731456}a^{29}-\frac{13723}{1595703296}a^{27}-\frac{8945}{99731456}a^{25}+\frac{335771}{1595703296}a^{23}-\frac{15823}{99731456}a^{21}-\frac{23299}{1595703296}a^{19}+\frac{113175}{99731456}a^{17}-\frac{2944773}{1595703296}a^{15}-\frac{571951}{99731456}a^{13}-\frac{740567}{49865728}a^{11}-\frac{144003}{6233216}a^{9}-\frac{12581}{779152}a^{7}-\frac{101399}{1558304}a^{5}+\frac{144}{48697}a^{3}+\frac{17895}{48697}a+1$, $\frac{35885}{1595703296}a^{31}+\frac{89}{12466432}a^{30}+\frac{55075}{1595703296}a^{27}-\frac{2309}{49865728}a^{26}+\frac{81373}{1595703296}a^{23}-\frac{9419}{49865728}a^{22}-\frac{182197}{1595703296}a^{19}-\frac{63765}{49865728}a^{18}+\frac{1409149}{1595703296}a^{15}-\frac{2835}{49865728}a^{14}+\frac{369}{779152}a^{11}-\frac{561457}{49865728}a^{10}-\frac{712715}{6233216}a^{7}+\frac{24327}{779152}a^{6}-\frac{32715}{389576}a^{3}+\frac{18933}{48697}a^{2}+1$, $\frac{149}{797851648}a^{31}+\frac{89}{12466432}a^{30}+\frac{93683}{797851648}a^{27}-\frac{2309}{49865728}a^{26}+\frac{151053}{797851648}a^{23}-\frac{9419}{49865728}a^{22}-\frac{59045}{797851648}a^{19}-\frac{63765}{49865728}a^{18}-\frac{3066515}{797851648}a^{15}-\frac{2835}{49865728}a^{14}-\frac{734313}{99731456}a^{11}-\frac{561457}{49865728}a^{10}-\frac{362231}{6233216}a^{7}+\frac{24327}{779152}a^{6}-\frac{644}{48697}a^{3}+\frac{18933}{48697}a^{2}+1$, $\frac{2593}{797851648}a^{31}-\frac{1269}{199462912}a^{29}+\frac{305}{9066496}a^{28}+\frac{34399}{797851648}a^{27}+\frac{16125}{199462912}a^{25}-\frac{9}{9066496}a^{24}-\frac{127199}{797851648}a^{23}-\frac{73213}{199462912}a^{21}-\frac{5879}{9066496}a^{20}-\frac{79449}{797851648}a^{19}-\frac{343147}{199462912}a^{17}-\frac{11201}{9066496}a^{16}+\frac{2176961}{797851648}a^{15}-\frac{497885}{199462912}a^{13}+\frac{120553}{9066496}a^{12}+\frac{764183}{49865728}a^{11}+\frac{1145307}{24932864}a^{9}-\frac{8109}{1133312}a^{8}-\frac{612067}{6233216}a^{7}-\frac{133345}{1558304}a^{5}-\frac{17337}{70832}a^{4}-\frac{33867}{389576}a^{3}-\frac{11595}{97394}a+\frac{2430}{4427}$, $\frac{1}{32768}a^{31}+\frac{5923}{398925824}a^{30}-\frac{1779}{199462912}a^{29}-\frac{1061}{99731456}a^{28}-\frac{1}{32768}a^{27}-\frac{34723}{398925824}a^{26}+\frac{16707}{199462912}a^{25}-\frac{19067}{99731456}a^{24}+\frac{1}{32768}a^{23}+\frac{292643}{398925824}a^{22}-\frac{1553}{10498048}a^{21}-\frac{107909}{99731456}a^{20}-\frac{57}{32768}a^{19}-\frac{949963}{398925824}a^{18}+\frac{393323}{199462912}a^{17}+\frac{492957}{99731456}a^{16}+\frac{161}{32768}a^{15}+\frac{1876227}{398925824}a^{14}-\frac{1619107}{199462912}a^{13}-\frac{268709}{99731456}a^{12}-\frac{57}{2048}a^{11}-\frac{453515}{24932864}a^{10}+\frac{4417}{97394}a^{9}+\frac{6913}{328064}a^{8}+\frac{1}{128}a^{7}+\frac{316933}{3116608}a^{6}-\frac{83013}{389576}a^{5}-\frac{31851}{389576}a^{4}-\frac{1}{8}a^{3}-\frac{12451}{48697}a^{2}+\frac{45857}{97394}a+\frac{61398}{48697}$, $\frac{355}{398925824}a^{31}-\frac{30699}{797851648}a^{30}-\frac{751}{18132992}a^{28}-\frac{7471}{398925824}a^{27}+\frac{13723}{797851648}a^{26}+\frac{4783}{18132992}a^{24}+\frac{67183}{398925824}a^{23}-\frac{335771}{797851648}a^{22}-\frac{23727}{18132992}a^{20}-\frac{7277}{20996096}a^{19}+\frac{23299}{797851648}a^{18}+\frac{65783}{18132992}a^{16}+\frac{85813}{20996096}a^{15}+\frac{2944773}{797851648}a^{14}-\frac{66575}{18132992}a^{12}-\frac{1700047}{99731456}a^{11}+\frac{740567}{24932864}a^{10}+\frac{67411}{1133312}a^{8}+\frac{141699}{1558304}a^{7}+\frac{12581}{389576}a^{6}-\frac{9797}{70832}a^{4}-\frac{83723}{194788}a^{3}-\frac{288}{48697}a^{2}-a+\frac{637}{4427}$, $\frac{35885}{1595703296}a^{31}-\frac{9501}{797851648}a^{30}-\frac{633}{49865728}a^{29}+\frac{3601}{199462912}a^{28}+\frac{55075}{1595703296}a^{27}-\frac{127283}{797851648}a^{26}+\frac{14183}{49865728}a^{25}-\frac{58497}{199462912}a^{24}+\frac{81373}{1595703296}a^{23}+\frac{122931}{797851648}a^{22}-\frac{903}{49865728}a^{21}-\frac{101247}{199462912}a^{20}-\frac{182197}{1595703296}a^{19}-\frac{971227}{797851648}a^{18}+\frac{76799}{49865728}a^{17}-\frac{423993}{199462912}a^{16}+\frac{1409149}{1595703296}a^{15}+\frac{2121619}{797851648}a^{14}-\frac{210231}{49865728}a^{13}-\frac{800863}{199462912}a^{12}+\frac{369}{779152}a^{11}-\frac{4515}{389576}a^{10}-\frac{763665}{24932864}a^{9}+\frac{41655}{779152}a^{8}-\frac{712715}{6233216}a^{7}+\frac{214783}{3116608}a^{6}-\frac{7023}{194788}a^{5}-\frac{5580}{48697}a^{4}-\frac{32715}{389576}a^{3}+\frac{5156}{48697}a^{2}-\frac{61397}{97394}a+\frac{42796}{48697}$, $\frac{355}{398925824}a^{31}-\frac{1169}{99731456}a^{30}+\frac{1169}{49865728}a^{28}-\frac{7471}{398925824}a^{27}+\frac{8945}{99731456}a^{26}-\frac{8945}{49865728}a^{24}+\frac{67183}{398925824}a^{23}+\frac{15823}{99731456}a^{22}-\frac{15823}{49865728}a^{20}-\frac{7277}{20996096}a^{19}-\frac{113175}{99731456}a^{18}+\frac{113175}{49865728}a^{16}+\frac{85813}{20996096}a^{15}+\frac{571951}{99731456}a^{14}-\frac{571951}{49865728}a^{12}-\frac{1700047}{99731456}a^{11}+\frac{144003}{6233216}a^{10}-\frac{144003}{3116608}a^{8}+\frac{141699}{1558304}a^{7}+\frac{101399}{1558304}a^{6}-\frac{101399}{779152}a^{4}-\frac{83723}{194788}a^{3}-\frac{17895}{48697}a^{2}+\frac{35790}{48697}$, $\frac{89}{12466432}a^{31}+\frac{5923}{398925824}a^{30}+\frac{89}{6233216}a^{29}-\frac{2309}{49865728}a^{27}-\frac{34723}{398925824}a^{26}-\frac{2309}{24932864}a^{25}-\frac{9419}{49865728}a^{23}+\frac{292643}{398925824}a^{22}-\frac{9419}{24932864}a^{21}-\frac{63765}{49865728}a^{19}-\frac{949963}{398925824}a^{18}-\frac{63765}{24932864}a^{17}-\frac{2835}{49865728}a^{15}+\frac{1876227}{398925824}a^{14}-\frac{2835}{24932864}a^{13}-\frac{561457}{49865728}a^{11}-\frac{453515}{24932864}a^{10}-\frac{561457}{24932864}a^{9}+\frac{24327}{779152}a^{7}+\frac{316933}{3116608}a^{6}+\frac{24327}{389576}a^{5}+\frac{18933}{48697}a^{3}-\frac{12451}{48697}a^{2}+\frac{37866}{48697}a$, $\frac{16883}{1595703296}a^{31}+\frac{1649}{49865728}a^{30}+\frac{1069}{199462912}a^{29}-\frac{7}{954368}a^{28}-\frac{199491}{1595703296}a^{27}-\frac{4513}{49865728}a^{26}-\frac{1765}{199462912}a^{25}+\frac{263}{954368}a^{24}+\frac{327235}{1595703296}a^{23}+\frac{12769}{49865728}a^{22}-\frac{104859}{199462912}a^{21}-\frac{519}{954368}a^{20}-\frac{2124651}{1595703296}a^{19}-\frac{72089}{49865728}a^{18}-\frac{116797}{199462912}a^{17}+\frac{911}{954368}a^{16}+\frac{5652387}{1595703296}a^{15}+\frac{220673}{49865728}a^{14}-\frac{1641787}{199462912}a^{13}-\frac{11879}{954368}a^{12}-\frac{8661}{779152}a^{11}-\frac{2073}{194788}a^{10}+\frac{569295}{24932864}a^{9}-\frac{465}{59648}a^{8}-\frac{283149}{6233216}a^{7}-\frac{124483}{779152}a^{6}-\frac{29343}{194788}a^{5}-\frac{87}{466}a^{4}+\frac{8533}{389576}a^{3}-\frac{12091}{194788}a^{2}+\frac{24195}{97394}a-\frac{89}{233}$, $\frac{6511}{1595703296}a^{31}-\frac{8771}{398925824}a^{30}-\frac{20893}{398925824}a^{29}+\frac{1225}{99731456}a^{28}-\frac{126767}{1595703296}a^{27}+\frac{53195}{398925824}a^{26}-\frac{53395}{398925824}a^{25}-\frac{17401}{99731456}a^{24}+\frac{541231}{1595703296}a^{23}-\frac{217291}{398925824}a^{22}+\frac{424339}{398925824}a^{21}-\frac{67719}{99731456}a^{20}-\frac{1557111}{1595703296}a^{19}+\frac{1460083}{398925824}a^{18}+\frac{1069253}{398925824}a^{17}+\frac{486543}{99731456}a^{16}+\frac{7985551}{1595703296}a^{15}-\frac{1853547}{398925824}a^{14}+\frac{2212339}{398925824}a^{13}+\frac{2699161}{99731456}a^{12}+\frac{1519}{5249024}a^{11}+\frac{1468487}{49865728}a^{10}+\frac{272497}{12466432}a^{9}-\frac{49475}{1558304}a^{8}+\frac{639623}{3116608}a^{7}-\frac{414241}{3116608}a^{6}+\frac{20167}{1558304}a^{5}-\frac{1035}{41008}a^{4}-\frac{102989}{389576}a^{3}+\frac{42215}{48697}a^{2}-\frac{16723}{48697}a+\frac{15236}{48697}$, $\frac{16063}{797851648}a^{31}-\frac{7497}{398925824}a^{30}-\frac{35831}{398925824}a^{29}-\frac{3823}{24932864}a^{28}-\frac{212687}{797851648}a^{27}-\frac{34327}{398925824}a^{26}-\frac{21913}{398925824}a^{25}+\frac{4107}{24932864}a^{24}+\frac{592719}{797851648}a^{23}+\frac{551319}{398925824}a^{22}+\frac{449177}{398925824}a^{21}+\frac{26421}{24932864}a^{20}-\frac{10093}{41992192}a^{19}-\frac{457119}{398925824}a^{18}-\frac{597137}{398925824}a^{17}-\frac{81645}{24932864}a^{16}+\frac{292085}{41992192}a^{15}-\frac{3428105}{398925824}a^{14}-\frac{1845319}{398925824}a^{13}-\frac{209483}{24932864}a^{12}-\frac{123779}{3116608}a^{11}-\frac{275117}{24932864}a^{10}+\frac{941627}{12466432}a^{9}+\frac{848435}{6233216}a^{8}+\frac{962681}{6233216}a^{7}+\frac{1079761}{3116608}a^{6}+\frac{466943}{1558304}a^{5}+\frac{32693}{194788}a^{4}-\frac{135103}{194788}a^{3}-\frac{39181}{48697}a^{2}-\frac{155953}{97394}a-\frac{99127}{48697}$
| 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: | \( 3273323441082.3687 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 3273323441082.3687 \cdot 150}{8\cdot\sqrt{3887187097015424555287785519115204034560000000000000000}}\cr\approx \mathstrut & 0.183675133490643 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 - x^28 + x^24 - 57*x^20 + 161*x^16 - 912*x^12 + 256*x^8 - 4096*x^4 + 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 - x^28 + x^24 - 57*x^20 + 161*x^16 - 912*x^12 + 256*x^8 - 4096*x^4 + 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 - x^28 + x^24 - 57*x^20 + 161*x^16 - 912*x^12 + 256*x^8 - 4096*x^4 + 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 - x^28 + x^24 - 57*x^20 + 161*x^16 - 912*x^12 + 256*x^8 - 4096*x^4 + 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 | ${\href{/padicField/3.8.0.1}{8} }^{4}$ | 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 | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $4$ | $4$ | $32$ | |||
| Deg $16$ | $4$ | $4$ | $32$ | ||||
|
\(5\)
| 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 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(1289\)
| $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |