Normalized defining polynomial
\( x^{22} - 11 x^{20} + 99 x^{18} - 33 x^{16} - 726 x^{14} + 23474 x^{12} - 4096 x^{11} + 25894 x^{10} + 22528 x^{9} + 141086 x^{8} + 540672 x^{7} + 2309285 x^{6} + \cdots + 5643763 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-129045097915537907803232946851577207980032\) \(\medspace = -\,2^{57}\cdot 11^{23}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(73.90\) | 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: | \(2\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-22}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{4}-\frac{1}{4}$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{8}a+\frac{1}{8}$, $\frac{1}{16}a^{6}+\frac{1}{16}a^{4}-\frac{1}{16}a^{2}-\frac{1}{16}$, $\frac{1}{16}a^{7}-\frac{1}{16}a^{5}-\frac{1}{8}a^{4}-\frac{1}{16}a^{3}+\frac{1}{16}a+\frac{1}{8}$, $\frac{1}{32}a^{8}-\frac{1}{16}a^{4}+\frac{1}{32}$, $\frac{1}{64}a^{9}-\frac{1}{64}a^{8}-\frac{1}{32}a^{7}-\frac{1}{32}a^{6}-\frac{1}{16}a^{5}+\frac{1}{32}a^{3}+\frac{1}{32}a^{2}+\frac{3}{64}a+\frac{1}{64}$, $\frac{1}{128}a^{10}+\frac{1}{128}a^{8}-\frac{1}{64}a^{6}+\frac{7}{64}a^{4}+\frac{1}{128}a^{2}-\frac{15}{128}$, $\frac{1}{1408}a^{11}-\frac{1}{128}a^{9}+\frac{1}{64}a^{7}-\frac{3}{64}a^{5}-\frac{5}{128}a^{3}-\frac{1}{4}a^{2}+\frac{39}{128}a+\frac{13}{44}$, $\frac{1}{2816}a^{12}-\frac{1}{256}a^{8}-\frac{7}{256}a^{4}+\frac{3}{11}a+\frac{101}{256}$, $\frac{1}{5632}a^{13}-\frac{1}{5632}a^{12}-\frac{1}{2816}a^{11}-\frac{1}{256}a^{10}+\frac{1}{512}a^{9}+\frac{7}{512}a^{8}-\frac{1}{128}a^{7}-\frac{3}{128}a^{6}+\frac{5}{512}a^{5}-\frac{53}{512}a^{4}-\frac{27}{256}a^{3}-\frac{243}{2816}a^{2}-\frac{1219}{5632}a-\frac{645}{5632}$, $\frac{1}{11264}a^{14}-\frac{1}{11264}a^{12}+\frac{3}{1024}a^{10}-\frac{11}{1024}a^{8}+\frac{17}{1024}a^{6}+\frac{127}{1024}a^{4}+\frac{3}{44}a^{3}+\frac{73}{1024}a^{2}+\frac{19}{44}a-\frac{209}{1024}$, $\frac{1}{22528}a^{15}-\frac{1}{22528}a^{14}-\frac{1}{22528}a^{13}-\frac{3}{22528}a^{12}+\frac{1}{22528}a^{11}+\frac{5}{2048}a^{10}-\frac{11}{2048}a^{9}-\frac{9}{2048}a^{8}+\frac{17}{2048}a^{7}+\frac{31}{2048}a^{6}-\frac{65}{2048}a^{5}+\frac{2319}{22528}a^{4}+\frac{1091}{22528}a^{3}-\frac{2187}{22528}a^{2}-\frac{8123}{22528}a+\frac{10327}{22528}$, $\frac{1}{22528}a^{16}-\frac{1}{256}a^{10}-\frac{13}{1024}a^{8}+\frac{1}{128}a^{6}+\frac{3}{88}a^{5}-\frac{11}{256}a^{4}+\frac{63}{256}a^{2}-\frac{1}{8}a-\frac{305}{2048}$, $\frac{1}{495616}a^{17}-\frac{1}{495616}a^{16}-\frac{3}{247808}a^{15}-\frac{3}{247808}a^{14}-\frac{1}{247808}a^{13}+\frac{3}{247808}a^{12}-\frac{23}{247808}a^{11}+\frac{27}{22528}a^{10}+\frac{5}{704}a^{9}+\frac{119}{11264}a^{8}-\frac{443}{22528}a^{7}-\frac{7305}{247808}a^{6}+\frac{11265}{247808}a^{5}+\frac{5957}{247808}a^{4}-\frac{4317}{247808}a^{3}+\frac{16227}{247808}a^{2}-\frac{93633}{495616}a+\frac{83197}{495616}$, $\frac{1}{1982464}a^{18}+\frac{15}{1982464}a^{16}+\frac{1}{61952}a^{15}+\frac{5}{123904}a^{14}-\frac{1}{15488}a^{13}+\frac{17}{247808}a^{12}-\frac{9}{61952}a^{11}+\frac{29}{8192}a^{10}+\frac{9}{2816}a^{9}-\frac{1019}{90112}a^{8}+\frac{1311}{61952}a^{7}+\frac{277}{22528}a^{6}+\frac{105}{3872}a^{5}-\frac{6395}{123904}a^{4}-\frac{1303}{61952}a^{3}-\frac{463603}{1982464}a^{2}+\frac{6615}{30976}a-\frac{446805}{1982464}$, $\frac{1}{3964928}a^{19}-\frac{1}{3964928}a^{18}-\frac{1}{3964928}a^{17}+\frac{3}{360448}a^{16}-\frac{1}{123904}a^{15}+\frac{1}{61952}a^{14}-\frac{29}{495616}a^{13}-\frac{87}{495616}a^{12}+\frac{413}{1982464}a^{11}-\frac{199}{180224}a^{10}+\frac{973}{180224}a^{9}-\frac{20351}{1982464}a^{8}+\frac{9309}{495616}a^{7}-\frac{9457}{495616}a^{6}-\frac{173}{2816}a^{5}-\frac{12219}{123904}a^{4}+\frac{283133}{3964928}a^{3}+\frac{475235}{3964928}a^{2}-\frac{1418037}{3964928}a+\frac{975093}{3964928}$, $\frac{1}{55508992}a^{20}+\frac{3}{27754496}a^{19}-\frac{1}{6938624}a^{18}-\frac{1}{2523136}a^{17}-\frac{15}{7929856}a^{16}-\frac{9}{867328}a^{15}+\frac{151}{6938624}a^{14}-\frac{95}{3469312}a^{13}+\frac{347}{2523136}a^{12}-\frac{4905}{13877248}a^{11}-\frac{79}{57344}a^{10}-\frac{86587}{13877248}a^{9}-\frac{401253}{27754496}a^{8}-\frac{106017}{3469312}a^{7}-\frac{3545}{630784}a^{6}-\frac{2363}{108416}a^{5}+\frac{451867}{7929856}a^{4}-\frac{3284873}{27754496}a^{3}-\frac{9261}{247808}a^{2}+\frac{744971}{2523136}a-\frac{11498077}{55508992}$, $\frac{1}{26\!\cdots\!08}a^{21}+\frac{29332957}{26\!\cdots\!08}a^{20}+\frac{1379310087}{13\!\cdots\!04}a^{19}+\frac{108233505}{12\!\cdots\!64}a^{18}-\frac{9145271503}{26\!\cdots\!08}a^{17}-\frac{182926033075}{26\!\cdots\!08}a^{16}+\frac{72985534583}{33\!\cdots\!76}a^{15}-\frac{11763812079}{301030608879616}a^{14}+\frac{64293840679}{18\!\cdots\!72}a^{13}-\frac{1591522641235}{13\!\cdots\!04}a^{12}-\frac{32660940201}{946096199335936}a^{11}+\frac{12704392989469}{66\!\cdots\!52}a^{10}+\frac{68703102746933}{13\!\cdots\!04}a^{9}+\frac{31537558968001}{13\!\cdots\!04}a^{8}-\frac{82974342819}{3909488427008}a^{7}+\frac{24218622574535}{33\!\cdots\!76}a^{6}+\frac{822153941009917}{26\!\cdots\!08}a^{5}-\frac{24\!\cdots\!03}{26\!\cdots\!08}a^{4}+\frac{118366184568845}{12\!\cdots\!64}a^{3}+\frac{691414248872275}{13\!\cdots\!04}a^{2}-\frac{12\!\cdots\!51}{26\!\cdots\!08}a-\frac{4863114693803}{19521513324544}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
Rank: | $10$ | 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: | $\frac{2399637741}{26\!\cdots\!08}a^{21}+\frac{4681511547}{26\!\cdots\!08}a^{20}-\frac{3925090341}{18\!\cdots\!72}a^{19}-\frac{4728503903}{18\!\cdots\!72}a^{18}+\frac{54495938379}{24\!\cdots\!28}a^{17}+\frac{571381036183}{26\!\cdots\!08}a^{16}-\frac{454122290157}{33\!\cdots\!76}a^{15}-\frac{178518637835}{33\!\cdots\!76}a^{14}+\frac{3514763349261}{13\!\cdots\!04}a^{13}+\frac{891303253051}{13\!\cdots\!04}a^{12}+\frac{12225555872567}{66\!\cdots\!52}a^{11}+\frac{10811014748333}{66\!\cdots\!52}a^{10}-\frac{280577052124851}{13\!\cdots\!04}a^{9}+\frac{83894180874723}{13\!\cdots\!04}a^{8}+\frac{95825830806163}{33\!\cdots\!76}a^{7}+\frac{6558823165511}{301030608879616}a^{6}+\frac{12\!\cdots\!21}{26\!\cdots\!08}a^{5}-\frac{70\!\cdots\!49}{26\!\cdots\!08}a^{4}-\frac{17\!\cdots\!03}{13\!\cdots\!04}a^{3}+\frac{66\!\cdots\!79}{13\!\cdots\!04}a^{2}-\frac{86\!\cdots\!55}{26\!\cdots\!08}a+\frac{63935659495535}{19521513324544}$, $\frac{8042064633}{37\!\cdots\!44}a^{21}+\frac{26925498267}{26\!\cdots\!08}a^{20}-\frac{306609091507}{13\!\cdots\!04}a^{19}-\frac{154330537107}{13\!\cdots\!04}a^{18}+\frac{5507605841399}{26\!\cdots\!08}a^{17}+\frac{400111425853}{37\!\cdots\!44}a^{16}-\frac{166787444319}{33\!\cdots\!76}a^{15}-\frac{278729637979}{33\!\cdots\!76}a^{14}-\frac{20645890862721}{13\!\cdots\!04}a^{13}-\frac{10039089933461}{13\!\cdots\!04}a^{12}+\frac{328463386031059}{66\!\cdots\!52}a^{11}+\frac{101357629141787}{66\!\cdots\!52}a^{10}+\frac{726325723651091}{13\!\cdots\!04}a^{9}+\frac{775306047756215}{13\!\cdots\!04}a^{8}+\frac{10\!\cdots\!13}{33\!\cdots\!76}a^{7}+\frac{41\!\cdots\!81}{33\!\cdots\!76}a^{6}+\frac{14\!\cdots\!11}{26\!\cdots\!08}a^{5}+\frac{10\!\cdots\!25}{37\!\cdots\!44}a^{4}+\frac{16\!\cdots\!49}{13\!\cdots\!04}a^{3}-\frac{22\!\cdots\!41}{18\!\cdots\!72}a^{2}+\frac{41\!\cdots\!15}{26\!\cdots\!08}a-\frac{413859198031837}{19521513324544}$, $\frac{18824011225}{13\!\cdots\!04}a^{21}+\frac{2907492667}{602061217759232}a^{20}-\frac{109220039263}{66\!\cdots\!52}a^{19}-\frac{198410780039}{33\!\cdots\!76}a^{18}+\frac{292620273971}{18\!\cdots\!72}a^{17}+\frac{3693010704917}{66\!\cdots\!52}a^{16}-\frac{326093980437}{16\!\cdots\!88}a^{15}-\frac{722009272403}{827834174418944}a^{14}-\frac{2693318069271}{66\!\cdots\!52}a^{13}-\frac{864358436397}{301030608879616}a^{12}+\frac{107484494211803}{33\!\cdots\!76}a^{11}+\frac{185900943315063}{16\!\cdots\!88}a^{10}-\frac{3139508030677}{602061217759232}a^{9}+\frac{9907362069313}{33\!\cdots\!76}a^{8}+\frac{680074733856451}{16\!\cdots\!88}a^{7}+\frac{11\!\cdots\!45}{827834174418944}a^{6}+\frac{10\!\cdots\!31}{18\!\cdots\!72}a^{5}+\frac{68\!\cdots\!49}{66\!\cdots\!52}a^{4}+\frac{47\!\cdots\!15}{946096199335936}a^{3}+\frac{13\!\cdots\!93}{33\!\cdots\!76}a^{2}-\frac{32\!\cdots\!57}{12\!\cdots\!64}a-\frac{8121958765301}{697196904448}$, $\frac{15382545917}{37\!\cdots\!44}a^{21}-\frac{19567397097}{26\!\cdots\!08}a^{20}-\frac{671622264175}{13\!\cdots\!04}a^{19}+\frac{154907825357}{13\!\cdots\!04}a^{18}+\frac{12579740977779}{26\!\cdots\!08}a^{17}-\frac{447366643015}{37\!\cdots\!44}a^{16}-\frac{2674238565755}{33\!\cdots\!76}a^{15}+\frac{1463757075433}{33\!\cdots\!76}a^{14}-\frac{26240203140517}{13\!\cdots\!04}a^{13}-\frac{1995023502905}{13\!\cdots\!04}a^{12}+\frac{650470023547247}{66\!\cdots\!52}a^{11}-\frac{243384111983261}{66\!\cdots\!52}a^{10}-\frac{411311100596225}{13\!\cdots\!04}a^{9}+\frac{25\!\cdots\!39}{13\!\cdots\!04}a^{8}+\frac{17\!\cdots\!73}{33\!\cdots\!76}a^{7}+\frac{61\!\cdots\!77}{33\!\cdots\!76}a^{6}+\frac{21\!\cdots\!47}{26\!\cdots\!08}a^{5}-\frac{11\!\cdots\!39}{37\!\cdots\!44}a^{4}+\frac{16\!\cdots\!57}{13\!\cdots\!04}a^{3}-\frac{47\!\cdots\!89}{18\!\cdots\!72}a^{2}+\frac{48\!\cdots\!55}{26\!\cdots\!08}a+\frac{150034654393895}{19521513324544}$, $\frac{84525090535}{26\!\cdots\!08}a^{21}-\frac{27484265525}{26\!\cdots\!08}a^{20}-\frac{229542191843}{13\!\cdots\!04}a^{19}+\frac{282711686937}{13\!\cdots\!04}a^{18}+\frac{2716105175167}{26\!\cdots\!08}a^{17}-\frac{5531915655197}{26\!\cdots\!08}a^{16}+\frac{895055535295}{473048099667968}a^{15}+\frac{443181892291}{473048099667968}a^{14}-\frac{67786991579625}{13\!\cdots\!04}a^{13}+\frac{16190085829691}{13\!\cdots\!04}a^{12}+\frac{441327357348435}{66\!\cdots\!52}a^{11}-\frac{324935475038793}{66\!\cdots\!52}a^{10}+\frac{68\!\cdots\!83}{13\!\cdots\!04}a^{9}+\frac{388198146454169}{18\!\cdots\!72}a^{8}+\frac{13\!\cdots\!53}{33\!\cdots\!76}a^{7}+\frac{11\!\cdots\!39}{473048099667968}a^{6}+\frac{26\!\cdots\!19}{26\!\cdots\!08}a^{5}+\frac{21\!\cdots\!87}{26\!\cdots\!08}a^{4}+\frac{83\!\cdots\!93}{13\!\cdots\!04}a^{3}-\frac{13\!\cdots\!79}{13\!\cdots\!04}a^{2}+\frac{25\!\cdots\!23}{26\!\cdots\!08}a-\frac{17\!\cdots\!89}{19521513324544}$, $\frac{82951707}{24\!\cdots\!28}a^{21}+\frac{885338725}{24\!\cdots\!28}a^{20}-\frac{4769124051}{13\!\cdots\!04}a^{19}-\frac{53244607261}{13\!\cdots\!04}a^{18}+\frac{119290236861}{26\!\cdots\!08}a^{17}+\frac{832862639267}{26\!\cdots\!08}a^{16}-\frac{84476155161}{33\!\cdots\!76}a^{15}+\frac{47391285625}{33\!\cdots\!76}a^{14}+\frac{1943848936041}{13\!\cdots\!04}a^{13}-\frac{7742718476393}{13\!\cdots\!04}a^{12}-\frac{1396187971793}{66\!\cdots\!52}a^{11}+\frac{546742620645}{86008745394176}a^{10}-\frac{5177562414813}{12\!\cdots\!64}a^{9}+\frac{100773227065311}{13\!\cdots\!04}a^{8}+\frac{132541652764911}{33\!\cdots\!76}a^{7}+\frac{8465106202321}{33\!\cdots\!76}a^{6}+\frac{787100704831685}{26\!\cdots\!08}a^{5}+\frac{90\!\cdots\!95}{26\!\cdots\!08}a^{4}-\frac{74\!\cdots\!55}{13\!\cdots\!04}a^{3}+\frac{75\!\cdots\!91}{13\!\cdots\!04}a^{2}-\frac{24\!\cdots\!91}{26\!\cdots\!08}a+\frac{11441500970843}{19521513324544}$, $\frac{15703124403}{13\!\cdots\!04}a^{21}-\frac{246600173}{473048099667968}a^{20}-\frac{103035244803}{66\!\cdots\!52}a^{19}+\frac{30617112599}{33\!\cdots\!76}a^{18}+\frac{1994450602115}{13\!\cdots\!04}a^{17}-\frac{21425571799}{206958543604736}a^{16}-\frac{593532869095}{16\!\cdots\!88}a^{15}+\frac{121519186827}{206958543604736}a^{14}-\frac{1802514521901}{66\!\cdots\!52}a^{13}-\frac{267755340689}{150515304439808}a^{12}+\frac{100635675937651}{33\!\cdots\!76}a^{11}-\frac{25346643865661}{16\!\cdots\!88}a^{10}-\frac{36287658932919}{946096199335936}a^{9}+\frac{27615013861839}{206958543604736}a^{8}+\frac{266404093812421}{16\!\cdots\!88}a^{7}+\frac{93212895976945}{413917087209472}a^{6}+\frac{33\!\cdots\!23}{13\!\cdots\!04}a^{5}-\frac{41\!\cdots\!23}{33\!\cdots\!76}a^{4}+\frac{14\!\cdots\!21}{66\!\cdots\!52}a^{3}-\frac{89\!\cdots\!09}{33\!\cdots\!76}a^{2}+\frac{12238508130203}{172017490788352}a+\frac{384353346173}{305023645696}$, $\frac{24852128453}{827834174418944}a^{21}+\frac{34187880631}{66\!\cdots\!52}a^{20}-\frac{2429147167347}{66\!\cdots\!52}a^{19}-\frac{363131421721}{66\!\cdots\!52}a^{18}+\frac{3207785978813}{946096199335936}a^{17}+\frac{1757341850833}{33\!\cdots\!76}a^{16}-\frac{994893197181}{206958543604736}a^{15}-\frac{554072497235}{827834174418944}a^{14}-\frac{15316794992927}{827834174418944}a^{13}+\frac{11560453025603}{33\!\cdots\!76}a^{12}+\frac{218060232462283}{301030608879616}a^{11}-\frac{116486173840097}{33\!\cdots\!76}a^{10}-\frac{212984654710501}{33\!\cdots\!76}a^{9}+\frac{17\!\cdots\!59}{16\!\cdots\!88}a^{8}+\frac{29\!\cdots\!05}{827834174418944}a^{7}+\frac{71\!\cdots\!11}{413917087209472}a^{6}+\frac{79\!\cdots\!83}{118262024916992}a^{5}-\frac{22\!\cdots\!45}{66\!\cdots\!52}a^{4}+\frac{96\!\cdots\!23}{946096199335936}a^{3}-\frac{11\!\cdots\!17}{66\!\cdots\!52}a^{2}-\frac{53\!\cdots\!69}{66\!\cdots\!52}a+\frac{3346716876301}{31690768384}$, $\frac{32623917783}{26\!\cdots\!08}a^{21}+\frac{1900883381513}{26\!\cdots\!08}a^{20}+\frac{424321734385}{13\!\cdots\!04}a^{19}-\frac{13569940968665}{13\!\cdots\!04}a^{18}-\frac{16426916533817}{26\!\cdots\!08}a^{17}+\frac{278171626953321}{26\!\cdots\!08}a^{16}+\frac{27545464230609}{33\!\cdots\!76}a^{15}-\frac{123127503713633}{33\!\cdots\!76}a^{14}-\frac{595310189571257}{13\!\cdots\!04}a^{13}+\frac{962909583912345}{13\!\cdots\!04}a^{12}+\frac{13\!\cdots\!15}{66\!\cdots\!52}a^{11}+\frac{13\!\cdots\!83}{946096199335936}a^{10}+\frac{432925410932899}{13\!\cdots\!04}a^{9}-\frac{42\!\cdots\!23}{13\!\cdots\!04}a^{8}+\frac{48\!\cdots\!57}{33\!\cdots\!76}a^{7}+\frac{81\!\cdots\!99}{33\!\cdots\!76}a^{6}+\frac{12\!\cdots\!75}{26\!\cdots\!08}a^{5}+\frac{28\!\cdots\!97}{26\!\cdots\!08}a^{4}-\frac{61\!\cdots\!31}{13\!\cdots\!04}a^{3}+\frac{64\!\cdots\!67}{13\!\cdots\!04}a^{2}-\frac{38\!\cdots\!05}{26\!\cdots\!08}a+\frac{11\!\cdots\!65}{19521513324544}$, $\frac{52000861959}{66\!\cdots\!52}a^{21}+\frac{67651105897}{66\!\cdots\!52}a^{20}-\frac{368043493425}{33\!\cdots\!76}a^{19}-\frac{48041480275}{301030608879616}a^{18}+\frac{7471957134291}{66\!\cdots\!52}a^{17}+\frac{11551969132801}{66\!\cdots\!52}a^{16}-\frac{448026839039}{118262024916992}a^{15}-\frac{87425479115}{10751093174272}a^{14}+\frac{19761588626959}{33\!\cdots\!76}a^{13}+\frac{90803456973681}{33\!\cdots\!76}a^{12}+\frac{282152490815753}{16\!\cdots\!88}a^{11}+\frac{172647485205997}{16\!\cdots\!88}a^{10}-\frac{13\!\cdots\!57}{33\!\cdots\!76}a^{9}-\frac{70822792146949}{473048099667968}a^{8}+\frac{208885484476313}{75257652219904}a^{7}+\frac{819416151372135}{118262024916992}a^{6}+\frac{95\!\cdots\!83}{66\!\cdots\!52}a^{5}-\frac{13\!\cdots\!75}{66\!\cdots\!52}a^{4}-\frac{10\!\cdots\!23}{301030608879616}a^{3}-\frac{49\!\cdots\!85}{33\!\cdots\!76}a^{2}-\frac{34\!\cdots\!21}{66\!\cdots\!52}a+\frac{58611071190073}{4880378331136}$ (assuming GRH) | 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: | \( 1485021311930 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 1485021311930 \cdot 2}{2\cdot\sqrt{129045097915537907803232946851577207980032}}\cr\approx \mathstrut & 2.49080932282076 \end{aligned}\] (assuming GRH)
Galois group
$M_{22}:C_2$ (as 22T41):
A non-solvable group of order 887040 |
The 21 conjugacy class representatives for $M_{22}:C_2$ |
Character table for $M_{22}:C_2$ is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 44 sibling: | 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 | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.11.0.1}{11} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.7.0.1}{7} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.5.0.1}{5} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.4.9.2 | $x^{4} + 2 x^{2} + 2$ | $4$ | $1$ | $9$ | $D_{4}$ | $[2, 3, 7/2]$ | |
2.8.22.24 | $x^{8} - 4 x^{6} + 16 x^{5} + 68 x^{4} - 32 x^{3} + 8 x^{2} + 416 x + 724$ | $4$ | $2$ | $22$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 2, 3, 7/2, 4]^{2}$ | |
2.8.26.40 | $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{4} + 8 x^{3} + 10$ | $8$ | $1$ | $26$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 2, 3, 7/2, 4]^{2}$ | |
\(11\) | Deg $22$ | $22$ | $1$ | $23$ |