Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 48*x^19 - 32*x^18 + 846*x^17 + 1128*x^16 - 5726*x^15 - 12204*x^14 - 3924*x^13 + 9424*x^12 + 186678*x^11 + 589812*x^10 - 71609*x^9 - 2888964*x^8 - 4329267*x^7 + 474058*x^6 + 8477028*x^5 + 11551176*x^4 + 8074864*x^3 + 3259872*x^2 + 724416*x + 68992)
gp: K = bnfinit(y^21 - 48*y^19 - 32*y^18 + 846*y^17 + 1128*y^16 - 5726*y^15 - 12204*y^14 - 3924*y^13 + 9424*y^12 + 186678*y^11 + 589812*y^10 - 71609*y^9 - 2888964*y^8 - 4329267*y^7 + 474058*y^6 + 8477028*y^5 + 11551176*y^4 + 8074864*y^3 + 3259872*y^2 + 724416*y + 68992, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 48*x^19 - 32*x^18 + 846*x^17 + 1128*x^16 - 5726*x^15 - 12204*x^14 - 3924*x^13 + 9424*x^12 + 186678*x^11 + 589812*x^10 - 71609*x^9 - 2888964*x^8 - 4329267*x^7 + 474058*x^6 + 8477028*x^5 + 11551176*x^4 + 8074864*x^3 + 3259872*x^2 + 724416*x + 68992);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 48*x^19 - 32*x^18 + 846*x^17 + 1128*x^16 - 5726*x^15 - 12204*x^14 - 3924*x^13 + 9424*x^12 + 186678*x^11 + 589812*x^10 - 71609*x^9 - 2888964*x^8 - 4329267*x^7 + 474058*x^6 + 8477028*x^5 + 11551176*x^4 + 8074864*x^3 + 3259872*x^2 + 724416*x + 68992)
\( x^{21} - 48 x^{19} - 32 x^{18} + 846 x^{17} + 1128 x^{16} - 5726 x^{15} - 12204 x^{14} - 3924 x^{13} + \cdots + 68992 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[17, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(10322551559300386187871072331953642204421617893376\)
\(\medspace = 2^{14}\cdot 3^{22}\cdot 7^{4}\cdot 11^{2}\cdot 1601^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(215.77\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(11\), \(1601\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{1601}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{8}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{4}a^{11}+\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{8}a^{3}+\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{64}a^{16}+\frac{1}{32}a^{15}-\frac{5}{16}a^{14}-\frac{1}{8}a^{13}-\frac{9}{32}a^{12}+\frac{1}{16}a^{11}-\frac{3}{32}a^{10}+\frac{1}{8}a^{9}+\frac{3}{16}a^{8}+\frac{1}{8}a^{7}+\frac{3}{32}a^{6}+\frac{23}{64}a^{4}+\frac{5}{32}a^{3}-\frac{23}{64}a^{2}-\frac{1}{16}a-\frac{7}{16}$, $\frac{1}{512}a^{17}-\frac{3}{64}a^{15}-\frac{3}{16}a^{14}-\frac{97}{256}a^{13}-\frac{19}{64}a^{12}+\frac{121}{256}a^{11}+\frac{37}{128}a^{10}+\frac{63}{128}a^{9}-\frac{5}{32}a^{8}-\frac{101}{256}a^{7}-\frac{51}{128}a^{6}+\frac{151}{512}a^{5}-\frac{41}{128}a^{4}+\frac{21}{512}a^{3}+\frac{53}{256}a^{2}+\frac{43}{128}a-\frac{9}{64}$, $\frac{1}{315392}a^{18}-\frac{1}{2048}a^{17}+\frac{85}{39424}a^{16}+\frac{1209}{19712}a^{15}-\frac{1537}{157696}a^{14}+\frac{31495}{78848}a^{13}-\frac{2617}{22528}a^{12}+\frac{291}{616}a^{11}+\frac{17373}{78848}a^{10}-\frac{2301}{39424}a^{9}-\frac{34005}{157696}a^{8}-\frac{1513}{39424}a^{7}+\frac{133071}{315392}a^{6}+\frac{24643}{157696}a^{5}+\frac{67069}{315392}a^{4}+\frac{13801}{39424}a^{3}-\frac{119}{512}a^{2}+\frac{9}{64}a-\frac{7}{256}$, $\frac{1}{2523136}a^{19}-\frac{1}{630784}a^{18}+\frac{555}{630784}a^{17}+\frac{3}{2464}a^{16}-\frac{68561}{1261568}a^{15}-\frac{60369}{157696}a^{14}+\frac{101477}{1261568}a^{13}-\frac{6117}{630784}a^{12}-\frac{119299}{630784}a^{11}+\frac{15857}{157696}a^{10}-\frac{202317}{1261568}a^{9}+\frac{304839}{630784}a^{8}-\frac{667361}{2523136}a^{7}+\frac{64205}{157696}a^{6}-\frac{44445}{229376}a^{5}-\frac{29885}{1261568}a^{4}-\frac{32797}{315392}a^{3}-\frac{721}{2048}a^{2}+\frac{513}{2048}a+\frac{99}{1024}$, $\frac{1}{20185088}a^{20}+\frac{1}{10092544}a^{19}+\frac{5}{5046272}a^{18}-\frac{415}{2523136}a^{17}-\frac{10705}{10092544}a^{16}+\frac{21039}{720896}a^{15}-\frac{3236427}{10092544}a^{14}+\frac{664421}{2523136}a^{13}+\frac{299615}{5046272}a^{12}+\frac{42399}{360448}a^{11}-\frac{9147}{1441792}a^{10}-\frac{20521}{157696}a^{9}+\frac{7665863}{20185088}a^{8}-\frac{1684027}{10092544}a^{7}+\frac{9899937}{20185088}a^{6}+\frac{677571}{5046272}a^{5}-\frac{410129}{5046272}a^{4}+\frac{369}{28672}a^{3}+\frac{6299}{16384}a^{2}+\frac{339}{4096}a-\frac{1719}{4096}$
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
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $18$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{198090711675}{2883584}a^{20}-\frac{482492118483}{10092544}a^{19}-\frac{1497429965309}{458752}a^{18}+\frac{184944696957}{2523136}a^{17}+\frac{586031521927443}{10092544}a^{16}+\frac{187116216875957}{5046272}a^{15}-\frac{384578784335709}{917504}a^{14}-\frac{13\!\cdots\!83}{2523136}a^{13}+\frac{559474581459779}{5046272}a^{12}+\frac{14\!\cdots\!65}{2523136}a^{11}+\frac{12\!\cdots\!91}{10092544}a^{10}+\frac{10\!\cdots\!73}{315392}a^{9}-\frac{54\!\cdots\!45}{20185088}a^{8}-\frac{18\!\cdots\!63}{10092544}a^{7}-\frac{34\!\cdots\!35}{20185088}a^{6}+\frac{76\!\cdots\!03}{5046272}a^{5}+\frac{24\!\cdots\!51}{5046272}a^{4}+\frac{14\!\cdots\!99}{315392}a^{3}+\frac{38\!\cdots\!51}{16384}a^{2}+\frac{252819796040847}{4096}a+\frac{27895801407461}{4096}$, $\frac{81497898549}{2883584}a^{20}-\frac{197697460113}{10092544}a^{19}-\frac{616120803659}{458752}a^{18}+\frac{66703337703}{2523136}a^{17}+\frac{241131837039645}{10092544}a^{16}+\frac{77313845783311}{5046272}a^{15}-\frac{158226884877555}{917504}a^{14}-\frac{568696945799403}{2523136}a^{13}+\frac{228761602232277}{5046272}a^{12}+\frac{592784732550663}{2523136}a^{11}+\frac{51\!\cdots\!37}{10092544}a^{10}+\frac{16\!\cdots\!01}{1261568}a^{9}-\frac{22\!\cdots\!67}{20185088}a^{8}-\frac{74\!\cdots\!81}{10092544}a^{7}-\frac{14\!\cdots\!65}{20185088}a^{6}+\frac{31\!\cdots\!75}{5046272}a^{5}+\frac{98\!\cdots\!33}{5046272}a^{4}+\frac{30\!\cdots\!63}{157696}a^{3}+\frac{15\!\cdots\!97}{16384}a^{2}+\frac{104354659079559}{4096}a+\frac{11520960485695}{4096}$, $\frac{129140163}{262144}a^{20}-\frac{43046721}{131072}a^{19}-\frac{1535333049}{65536}a^{18}-\frac{4782969}{32768}a^{17}+\frac{54639043533}{131072}a^{16}+\frac{18204511455}{65536}a^{15}-\frac{394000968609}{131072}a^{14}-\frac{131336490555}{32768}a^{13}+\frac{48428820837}{65536}a^{12}+\frac{135984171735}{32768}a^{11}+\frac{11691189216297}{131072}a^{10}+\frac{3786260345685}{16384}a^{9}-\frac{49634374952907}{262144}a^{8}-\frac{169995849279597}{131072}a^{7}-\frac{332421113677725}{262144}a^{6}+\frac{70708500794151}{65536}a^{5}+\frac{226542193889457}{65536}a^{4}+\frac{6934439331587}{2048}a^{3}+\frac{28190651882085}{16384}a^{2}+\frac{1879376792139}{4096}a+\frac{208819651763}{4096}$, $\frac{5998190157}{90112}a^{20}-\frac{117191333367}{2523136}a^{19}-\frac{181359064699}{57344}a^{18}+\frac{48438997083}{630784}a^{17}+\frac{17743769142501}{315392}a^{16}+\frac{45198493858583}{1261568}a^{15}-\frac{11644986326661}{28672}a^{14}-\frac{667299623001243}{1261568}a^{13}+\frac{68062475734103}{630784}a^{12}+\frac{348200455718277}{630784}a^{11}+\frac{37\!\cdots\!19}{315392}a^{10}+\frac{38\!\cdots\!07}{1261568}a^{9}-\frac{41\!\cdots\!61}{157696}a^{8}-\frac{43\!\cdots\!13}{2523136}a^{7}-\frac{10\!\cdots\!17}{630784}a^{6}+\frac{37\!\cdots\!01}{2523136}a^{5}+\frac{58\!\cdots\!27}{1261568}a^{4}+\frac{14\!\cdots\!19}{315392}a^{3}+\frac{461717084888955}{2048}a^{2}+\frac{122205216361473}{2048}a+\frac{6738979177699}{1024}$, $\frac{5583411783}{65536}a^{20}-\frac{6810336081}{114688}a^{19}-\frac{464259663205}{114688}a^{18}+\frac{341133633}{3584}a^{17}+\frac{16517251718463}{229376}a^{16}+\frac{1316355825583}{28672}a^{15}-\frac{119236887239787}{229376}a^{14}-\frac{77691062928789}{114688}a^{13}+\frac{15808269234669}{114688}a^{12}+\frac{20265698661489}{28672}a^{11}+\frac{35\!\cdots\!35}{229376}a^{10}+\frac{45\!\cdots\!19}{114688}a^{9}-\frac{15\!\cdots\!29}{458752}a^{8}-\frac{63\!\cdots\!19}{28672}a^{7}-\frac{98\!\cdots\!03}{458752}a^{6}+\frac{43\!\cdots\!55}{229376}a^{5}+\frac{33\!\cdots\!91}{57344}a^{4}+\frac{16\!\cdots\!51}{28672}a^{3}+\frac{11\!\cdots\!19}{4096}a^{2}+\frac{156569539893921}{2048}a+\frac{539734756381}{64}$, $\frac{110444186475}{2883584}a^{20}-\frac{271067873355}{10092544}a^{19}-\frac{834751713933}{458752}a^{18}+\frac{127052650629}{2523136}a^{17}+\frac{326668775520387}{10092544}a^{16}+\frac{103480129651021}{5046272}a^{15}-\frac{214412677575597}{917504}a^{14}-\frac{765902300203083}{2523136}a^{13}+\frac{315738203640547}{5046272}a^{12}+\frac{800017090322445}{2523136}a^{11}+\frac{69\!\cdots\!71}{10092544}a^{10}+\frac{11\!\cdots\!03}{630784}a^{9}-\frac{30\!\cdots\!41}{20185088}a^{8}-\frac{10\!\cdots\!71}{10092544}a^{7}-\frac{19\!\cdots\!79}{20185088}a^{6}+\frac{43\!\cdots\!27}{5046272}a^{5}+\frac{13\!\cdots\!75}{5046272}a^{4}+\frac{80\!\cdots\!93}{315392}a^{3}+\frac{21\!\cdots\!43}{16384}a^{2}+\frac{140055773643387}{4096}a+\frac{15434956216909}{4096}$, $\frac{8769974017}{1441792}a^{20}-\frac{21329753553}{5046272}a^{19}-\frac{104181188171}{360448}a^{18}+\frac{7824418463}{1261568}a^{17}+\frac{25946281354313}{5046272}a^{16}+\frac{8296899392087}{2523136}a^{15}-\frac{187292417376781}{5046272}a^{14}-\frac{8730421971999}{180224}a^{13}+\frac{24719725781577}{2523136}a^{12}+\frac{5793723175813}{114688}a^{11}+\frac{55\!\cdots\!21}{5046272}a^{10}+\frac{890345028528099}{315392}a^{9}-\frac{24\!\cdots\!99}{10092544}a^{8}-\frac{80\!\cdots\!49}{5046272}a^{7}-\frac{14\!\cdots\!31}{917504}a^{6}+\frac{34\!\cdots\!81}{2523136}a^{5}+\frac{96\!\cdots\!19}{229376}a^{4}+\frac{922504467594993}{22528}a^{3}+\frac{169235748782605}{8192}a^{2}+\frac{11205251687913}{2048}a+\frac{1236583312695}{2048}$, $\frac{3487046545}{262144}a^{20}-\frac{1162392539}{131072}a^{19}-\frac{41457072515}{65536}a^{18}-\frac{128648643}{32768}a^{17}+\frac{1475363096223}{131072}a^{16}+\frac{491541273477}{65536}a^{15}-\frac{10638815599099}{131072}a^{14}-\frac{3546287783993}{32768}a^{13}+\frac{1307726077351}{65536}a^{12}+\frac{3671807485101}{32768}a^{11}+\frac{315685637705811}{131072}a^{10}+\frac{102235751705079}{16384}a^{9}-\frac{13\!\cdots\!29}{262144}a^{8}-\frac{45\!\cdots\!07}{131072}a^{7}-\frac{89\!\cdots\!47}{262144}a^{6}+\frac{19\!\cdots\!33}{65536}a^{5}+\frac{61\!\cdots\!91}{65536}a^{4}+\frac{187241065370961}{2048}a^{3}+\frac{761190272190631}{16384}a^{2}+\frac{50745857992169}{4096}a+\frac{5638412971745}{4096}$, $\frac{849747394193}{5046272}a^{20}-\frac{148090267233}{1261568}a^{19}-\frac{10093740051733}{1261568}a^{18}+\frac{7450010959}{39424}a^{17}+\frac{359111273247647}{2523136}a^{16}+\frac{28614972044871}{315392}a^{15}-\frac{235674204700673}{229376}a^{14}-\frac{21934922884713}{16384}a^{13}+\frac{343829300698925}{1261568}a^{12}+\frac{40053185902483}{28672}a^{11}+\frac{10\!\cdots\!33}{360448}a^{10}+\frac{98\!\cdots\!95}{1261568}a^{9}-\frac{30\!\cdots\!15}{458752}a^{8}-\frac{13\!\cdots\!15}{315392}a^{7}-\frac{30\!\cdots\!89}{720896}a^{6}+\frac{94\!\cdots\!59}{2523136}a^{5}+\frac{73\!\cdots\!27}{630784}a^{4}+\frac{32\!\cdots\!01}{28672}a^{3}+\frac{23\!\cdots\!69}{4096}a^{2}+\frac{309404963907859}{2048}a+\frac{1066544774613}{64}$, $\frac{21556061013}{10092544}a^{20}-\frac{7444614453}{5046272}a^{19}-\frac{256101369955}{2523136}a^{18}+\frac{202083733}{114688}a^{17}+\frac{1301720733133}{720896}a^{16}+\frac{266534003221}{229376}a^{15}-\frac{9394962078017}{720896}a^{14}-\frac{1345467173199}{78848}a^{13}+\frac{8590660199727}{2523136}a^{12}+\frac{22425451703467}{1261568}a^{11}+\frac{19\!\cdots\!63}{5046272}a^{10}+\frac{178930841052615}{180224}a^{9}-\frac{84\!\cdots\!73}{10092544}a^{8}-\frac{28\!\cdots\!61}{5046272}a^{7}-\frac{54\!\cdots\!51}{10092544}a^{6}+\frac{59\!\cdots\!09}{1261568}a^{5}+\frac{37\!\cdots\!73}{2523136}a^{4}+\frac{45\!\cdots\!39}{315392}a^{3}+\frac{59675791242059}{8192}a^{2}+\frac{1977689453301}{1024}a+\frac{436968470271}{2048}$, $\frac{470960943451}{10092544}a^{20}-\frac{165313069807}{5046272}a^{19}-\frac{5593501006925}{2523136}a^{18}+\frac{11363051107}{180224}a^{17}+\frac{198992802641205}{5046272}a^{16}+\frac{62962183401949}{2523136}a^{15}-\frac{205251083733519}{720896}a^{14}-\frac{21195054301395}{57344}a^{13}+\frac{192662358763225}{2523136}a^{12}+\frac{487158744357057}{1261568}a^{11}+\frac{42\!\cdots\!49}{5046272}a^{10}+\frac{24\!\cdots\!31}{114688}a^{9}-\frac{18\!\cdots\!55}{10092544}a^{8}-\frac{61\!\cdots\!67}{5046272}a^{7}-\frac{11\!\cdots\!49}{10092544}a^{6}+\frac{65\!\cdots\!91}{630784}a^{5}+\frac{81\!\cdots\!11}{2523136}a^{4}+\frac{98\!\cdots\!47}{315392}a^{3}+\frac{12\!\cdots\!09}{8192}a^{2}+\frac{21310385373323}{512}a+\frac{9392618387533}{2048}$, $\frac{42774551011}{20185088}a^{20}-\frac{15765731649}{10092544}a^{19}-\frac{507484438121}{5046272}a^{18}+\frac{15949554967}{2523136}a^{17}+\frac{18046672959661}{10092544}a^{16}+\frac{5410805965087}{5046272}a^{15}-\frac{11858357180899}{917504}a^{14}-\frac{41213610367947}{2523136}a^{13}+\frac{18804178831205}{5046272}a^{12}+\frac{43459079443799}{2523136}a^{11}+\frac{38\!\cdots\!33}{10092544}a^{10}+\frac{12\!\cdots\!45}{1261568}a^{9}-\frac{24\!\cdots\!93}{2883584}a^{8}-\frac{55\!\cdots\!09}{10092544}a^{7}-\frac{10\!\cdots\!13}{20185088}a^{6}+\frac{24\!\cdots\!03}{5046272}a^{5}+\frac{72\!\cdots\!93}{5046272}a^{4}+\frac{21\!\cdots\!27}{157696}a^{3}+\frac{113225742625913}{16384}a^{2}+\frac{7427120975815}{4096}a+\frac{812251158831}{4096}$, $\frac{3528246734367}{10092544}a^{20}-\frac{1211881112433}{5046272}a^{19}-\frac{41922796854749}{2523136}a^{18}+\frac{286708944583}{1261568}a^{17}+\frac{14\!\cdots\!37}{5046272}a^{16}+\frac{482604242080999}{2523136}a^{15}-\frac{10\!\cdots\!17}{5046272}a^{14}-\frac{35\!\cdots\!25}{1261568}a^{13}+\frac{13\!\cdots\!25}{2523136}a^{12}+\frac{36\!\cdots\!55}{1261568}a^{11}+\frac{31\!\cdots\!05}{5046272}a^{10}+\frac{25\!\cdots\!27}{157696}a^{9}-\frac{13\!\cdots\!91}{10092544}a^{8}-\frac{66\!\cdots\!55}{720896}a^{7}-\frac{89\!\cdots\!05}{10092544}a^{6}+\frac{19\!\cdots\!77}{2523136}a^{5}+\frac{87\!\cdots\!63}{360448}a^{4}+\frac{37\!\cdots\!85}{157696}a^{3}+\frac{98\!\cdots\!45}{8192}a^{2}+\frac{650234484744357}{2048}a+\frac{71889782415951}{2048}$, $\frac{21346708881}{315392}a^{20}-\frac{60031276947}{1261568}a^{19}-\frac{63379973375}{19712}a^{18}+\frac{4262774555}{45056}a^{17}+\frac{1288415388981}{22528}a^{16}+\frac{2072354383417}{57344}a^{15}-\frac{32558843117379}{78848}a^{14}-\frac{48272366763977}{90112}a^{13}+\frac{34950049339973}{315392}a^{12}+\frac{16050567489659}{28672}a^{11}+\frac{19\!\cdots\!27}{157696}a^{10}+\frac{17\!\cdots\!25}{57344}a^{9}-\frac{529381913372139}{19712}a^{8}-\frac{31\!\cdots\!87}{180224}a^{7}-\frac{13\!\cdots\!59}{78848}a^{6}+\frac{19\!\cdots\!41}{1261568}a^{5}+\frac{29\!\cdots\!45}{630784}a^{4}+\frac{71\!\cdots\!23}{157696}a^{3}+\frac{233653516122485}{1024}a^{2}+\frac{61786965501389}{1024}a+\frac{3404508874193}{512}$, $\frac{6444224675181}{1835008}a^{20}-\frac{24687730887321}{10092544}a^{19}-\frac{120291376862419}{720896}a^{18}+\frac{9712449058871}{2523136}a^{17}+\frac{29\!\cdots\!17}{10092544}a^{16}+\frac{95\!\cdots\!91}{5046272}a^{15}-\frac{30\!\cdots\!75}{1441792}a^{14}-\frac{70\!\cdots\!65}{2523136}a^{13}+\frac{28\!\cdots\!29}{5046272}a^{12}+\frac{73\!\cdots\!75}{2523136}a^{11}+\frac{64\!\cdots\!41}{10092544}a^{10}+\frac{10\!\cdots\!61}{630784}a^{9}-\frac{39\!\cdots\!17}{2883584}a^{8}-\frac{92\!\cdots\!53}{10092544}a^{7}-\frac{17\!\cdots\!37}{20185088}a^{6}+\frac{39\!\cdots\!85}{5046272}a^{5}+\frac{12\!\cdots\!09}{5046272}a^{4}+\frac{74\!\cdots\!75}{315392}a^{3}+\frac{19\!\cdots\!01}{16384}a^{2}+\frac{12\!\cdots\!45}{4096}a+\frac{14\!\cdots\!91}{4096}$, $\frac{339847594767}{10092544}a^{20}-\frac{28131382251}{720896}a^{19}-\frac{568804382139}{360448}a^{18}+\frac{960068661467}{1261568}a^{17}+\frac{12812623506147}{458752}a^{16}+\frac{13571829360515}{2523136}a^{15}-\frac{10\!\cdots\!69}{5046272}a^{14}-\frac{218688243246859}{1261568}a^{13}+\frac{280066582000553}{2523136}a^{12}+\frac{269994898317035}{1261568}a^{11}+\frac{30\!\cdots\!01}{5046272}a^{10}+\frac{81\!\cdots\!03}{630784}a^{9}-\frac{17\!\cdots\!97}{917504}a^{8}-\frac{56\!\cdots\!15}{720896}a^{7}-\frac{51\!\cdots\!61}{10092544}a^{6}+\frac{20\!\cdots\!49}{229376}a^{5}+\frac{47\!\cdots\!65}{2523136}a^{4}+\frac{59\!\cdots\!03}{39424}a^{3}+\frac{518750394544749}{8192}a^{2}+\frac{27922825888783}{2048}a+\frac{2391889695955}{2048}$, $\frac{3248869499}{28672}a^{20}-\frac{99713713745}{1261568}a^{19}-\frac{424503498953}{78848}a^{18}+\frac{40832216055}{315392}a^{17}+\frac{2157533895925}{22528}a^{16}+\frac{5497834723943}{90112}a^{15}-\frac{3407113666305}{4928}a^{14}-\frac{51643395534799}{57344}a^{13}+\frac{5262865806085}{28672}a^{12}+\frac{296407175972387}{315392}a^{11}+\frac{293848312728445}{14336}a^{10}+\frac{33\!\cdots\!29}{630784}a^{9}-\frac{70\!\cdots\!59}{157696}a^{8}-\frac{33\!\cdots\!21}{114688}a^{7}-\frac{63\!\cdots\!59}{22528}a^{6}+\frac{31\!\cdots\!91}{1261568}a^{5}+\frac{49\!\cdots\!79}{630784}a^{4}+\frac{12\!\cdots\!01}{157696}a^{3}+\frac{393096116977095}{1024}a^{2}+\frac{104050986086047}{1024}a+\frac{5738332885027}{512}$, $\frac{618349833421623}{20185088}a^{20}-\frac{215363619838809}{10092544}a^{19}-\frac{73\!\cdots\!81}{5046272}a^{18}+\frac{7712690060741}{229376}a^{17}+\frac{37\!\cdots\!43}{1441792}a^{16}+\frac{11\!\cdots\!25}{720896}a^{15}-\frac{18\!\cdots\!25}{10092544}a^{14}-\frac{61\!\cdots\!05}{2523136}a^{13}+\frac{24\!\cdots\!65}{5046272}a^{12}+\frac{64\!\cdots\!27}{2523136}a^{11}+\frac{55\!\cdots\!77}{10092544}a^{10}+\frac{44\!\cdots\!03}{315392}a^{9}-\frac{24\!\cdots\!63}{20185088}a^{8}-\frac{80\!\cdots\!21}{10092544}a^{7}-\frac{22\!\cdots\!55}{2883584}a^{6}+\frac{34\!\cdots\!53}{5046272}a^{5}+\frac{10\!\cdots\!05}{5046272}a^{4}+\frac{59\!\cdots\!15}{28672}a^{3}+\frac{17\!\cdots\!85}{16384}a^{2}+\frac{11\!\cdots\!41}{4096}a+\frac{12\!\cdots\!67}{4096}$
| 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: | \( 523529356224000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{17}\cdot(2\pi)^{2}\cdot 523529356224000000 \cdot 1}{2\cdot\sqrt{10322551559300386187871072331953642204421617893376}}\cr\approx \mathstrut & 0.421586966587329 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 48*x^19 - 32*x^18 + 846*x^17 + 1128*x^16 - 5726*x^15 - 12204*x^14 - 3924*x^13 + 9424*x^12 + 186678*x^11 + 589812*x^10 - 71609*x^9 - 2888964*x^8 - 4329267*x^7 + 474058*x^6 + 8477028*x^5 + 11551176*x^4 + 8074864*x^3 + 3259872*x^2 + 724416*x + 68992)
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^21 - 48*x^19 - 32*x^18 + 846*x^17 + 1128*x^16 - 5726*x^15 - 12204*x^14 - 3924*x^13 + 9424*x^12 + 186678*x^11 + 589812*x^10 - 71609*x^9 - 2888964*x^8 - 4329267*x^7 + 474058*x^6 + 8477028*x^5 + 11551176*x^4 + 8074864*x^3 + 3259872*x^2 + 724416*x + 68992, 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^21 - 48*x^19 - 32*x^18 + 846*x^17 + 1128*x^16 - 5726*x^15 - 12204*x^14 - 3924*x^13 + 9424*x^12 + 186678*x^11 + 589812*x^10 - 71609*x^9 - 2888964*x^8 - 4329267*x^7 + 474058*x^6 + 8477028*x^5 + 11551176*x^4 + 8074864*x^3 + 3259872*x^2 + 724416*x + 68992);
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^21 - 48*x^19 - 32*x^18 + 846*x^17 + 1128*x^16 - 5726*x^15 - 12204*x^14 - 3924*x^13 + 9424*x^12 + 186678*x^11 + 589812*x^10 - 71609*x^9 - 2888964*x^8 - 4329267*x^7 + 474058*x^6 + 8477028*x^5 + 11551176*x^4 + 8074864*x^3 + 3259872*x^2 + 724416*x + 68992);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_3^7:(C_2^6:D_7)$ (as 21T122):
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 1959552 |
| The 171 conjugacy class representatives for $C_3^7:(C_2^6:D_7)$ |
| Character table for $C_3^7:(C_2^6:D_7)$ |
Intermediate fields
| 7.7.4103684801.1 |
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 42 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | $21$ | R | R | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{5}$ | ${\href{/padicField/19.7.0.1}{7} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $21$ | $21$ | ${\href{/padicField/37.7.0.1}{7} }^{3}$ | ${\href{/padicField/41.7.0.1}{7} }^{3}$ | $21$ | $21$ | ${\href{/padicField/53.7.0.1}{7} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.7.0.1 | $x^{7} + x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 2.14.14.34 | $x^{14} + 14 x^{13} + 98 x^{12} + 680 x^{11} + 4124 x^{10} + 16872 x^{9} + 59992 x^{8} + 168192 x^{7} + 397680 x^{6} + 807456 x^{5} + 1341920 x^{4} + 1886592 x^{3} + 2248768 x^{2} + 1722752 x + 1571456$ | $2$ | $7$ | $14$ | 14T21 | $[2, 2, 2, 2, 2, 2]^{7}$ | |
|
\(3\)
| 3.3.4.3 | $x^{3} + 6 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
| 3.6.6.2 | $x^{6} - 6 x^{5} + 39 x^{4} + 60 x^{3} - 18 x + 9$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
| 3.6.6.5 | $x^{6} + 6 x^{3} + 36 x^{2} + 9$ | $3$ | $2$ | $6$ | $S_3^2$ | $[3/2, 3/2]_{2}^{2}$ | |
| 3.6.6.1 | $x^{6} - 6 x^{5} + 24 x^{4} + 6 x^{3} + 18 x + 9$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.6.4.1 | $x^{6} + 14 x^{3} - 245$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(11\)
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.3.2.1 | $x^{3} + 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.6.0.1 | $x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(1601\)
| $\Q_{1601}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1601}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1601}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $2$ | $3$ | $3$ |