Normalized defining polynomial
\( x^{24} - 2 x^{23} + 2 x^{22} + 10 x^{21} - 20 x^{20} + 13 x^{19} + 57 x^{18} - 98 x^{17} + 19 x^{16} + 228 x^{15} - 267 x^{14} - 159 x^{13} + 711 x^{12} - 318 x^{11} - 1068 x^{10} + 1824 x^{9} + \cdots + 4096 \)
Invariants
Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(138359014736314946502328332753681\)
\(\medspace = 3^{12}\cdot 7^{20}\cdot 239^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(21.84\) | 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))
| |
Ramified primes: |
\(3\), \(7\), \(239\)
| 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$^{2048}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{10}+\frac{1}{8}a^{9}+\frac{1}{4}a^{8}+\frac{1}{8}a^{7}+\frac{1}{4}a^{6}+\frac{1}{8}a^{5}-\frac{1}{8}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{16}-\frac{1}{8}a^{14}-\frac{1}{8}a^{13}-\frac{3}{16}a^{11}+\frac{3}{16}a^{10}-\frac{1}{16}a^{8}+\frac{1}{8}a^{7}+\frac{5}{16}a^{6}-\frac{5}{16}a^{5}-\frac{7}{16}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{32}a^{17}-\frac{1}{16}a^{15}-\frac{1}{16}a^{14}-\frac{3}{32}a^{12}-\frac{5}{32}a^{11}+\frac{7}{32}a^{9}-\frac{7}{16}a^{8}-\frac{11}{32}a^{7}+\frac{3}{32}a^{6}-\frac{7}{32}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{64}a^{18}-\frac{1}{32}a^{16}-\frac{1}{32}a^{15}+\frac{5}{64}a^{13}+\frac{3}{64}a^{12}-\frac{1}{64}a^{10}+\frac{5}{32}a^{9}-\frac{19}{64}a^{8}-\frac{21}{64}a^{7}+\frac{25}{64}a^{6}-\frac{3}{8}a^{4}-\frac{1}{8}a^{3}$, $\frac{1}{128}a^{19}-\frac{1}{64}a^{17}-\frac{1}{64}a^{16}+\frac{5}{128}a^{14}-\frac{13}{128}a^{13}-\frac{1}{8}a^{12}+\frac{31}{128}a^{11}+\frac{13}{64}a^{10}+\frac{29}{128}a^{9}+\frac{59}{128}a^{8}+\frac{9}{128}a^{7}+\frac{1}{4}a^{6}-\frac{3}{16}a^{5}-\frac{1}{16}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{512}a^{20}-\frac{1}{256}a^{19}-\frac{1}{256}a^{18}+\frac{1}{256}a^{17}+\frac{1}{128}a^{16}+\frac{5}{512}a^{15}+\frac{41}{512}a^{14}+\frac{21}{256}a^{13}-\frac{33}{512}a^{12}+\frac{23}{128}a^{11}-\frac{119}{512}a^{10}+\frac{33}{512}a^{9}-\frac{141}{512}a^{8}-\frac{41}{256}a^{7}-\frac{3}{64}a^{6}+\frac{29}{64}a^{5}-\frac{1}{8}a^{4}+\frac{1}{8}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{1024}a^{21}+\frac{1}{512}a^{19}-\frac{1}{512}a^{18}-\frac{1}{128}a^{17}-\frac{3}{1024}a^{16}+\frac{51}{1024}a^{15}+\frac{9}{256}a^{14}+\frac{75}{1024}a^{13}-\frac{51}{512}a^{12}+\frac{185}{1024}a^{11}-\frac{125}{1024}a^{10}-\frac{227}{1024}a^{9}+\frac{123}{256}a^{8}-\frac{29}{256}a^{7}-\frac{41}{128}a^{6}+\frac{29}{64}a^{5}+\frac{7}{16}a^{4}-\frac{5}{16}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{176128}a^{22}+\frac{19}{44032}a^{21}+\frac{83}{88064}a^{20}+\frac{39}{88064}a^{19}-\frac{165}{22016}a^{18}+\frac{613}{176128}a^{17}-\frac{2465}{176128}a^{16}+\frac{127}{44032}a^{15}+\frac{4927}{176128}a^{14}-\frac{10685}{88064}a^{13}-\frac{11763}{176128}a^{12}-\frac{41249}{176128}a^{11}+\frac{12293}{176128}a^{10}+\frac{8171}{44032}a^{9}-\frac{319}{22016}a^{8}+\frac{343}{1376}a^{7}-\frac{5403}{11008}a^{6}+\frac{2301}{5504}a^{5}+\frac{155}{2752}a^{4}-\frac{575}{1376}a^{3}-\frac{3}{172}a^{2}+\frac{109}{344}a+\frac{45}{172}$, $\frac{1}{44736512}a^{23}+\frac{7}{22368256}a^{22}-\frac{6745}{22368256}a^{21}-\frac{17491}{22368256}a^{20}-\frac{10139}{11184128}a^{19}+\frac{28789}{44736512}a^{18}-\frac{656919}{44736512}a^{17}+\frac{7599}{520192}a^{16}-\frac{2742105}{44736512}a^{15}+\frac{789297}{11184128}a^{14}-\frac{39205}{1040384}a^{13}-\frac{2106599}{44736512}a^{12}+\frac{67475}{44736512}a^{11}-\frac{2905869}{22368256}a^{10}-\frac{305561}{2796032}a^{9}-\frac{1375407}{2796032}a^{8}+\frac{714569}{2796032}a^{7}-\frac{303565}{699008}a^{6}+\frac{15273}{174752}a^{5}+\frac{741}{5461}a^{4}-\frac{42903}{174752}a^{3}+\frac{33849}{87376}a^{2}-\frac{7859}{21844}a+\frac{2905}{21844}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: |
\( \frac{233629}{22368256} a^{23} + \frac{356283}{22368256} a^{22} - \frac{244523}{11184128} a^{21} + \frac{59507}{699008} a^{20} + \frac{1973597}{11184128} a^{19} - \frac{5059799}{22368256} a^{18} + \frac{2173295}{11184128} a^{17} + \frac{22093205}{22368256} a^{16} - \frac{502931}{520192} a^{15} - \frac{13893057}{22368256} a^{14} + \frac{77191467}{22368256} a^{13} - \frac{16518885}{11184128} a^{12} - \frac{59569811}{11184128} a^{11} + \frac{181994879}{22368256} a^{10} + \frac{23259973}{5592064} a^{9} - \frac{43889855}{2796032} a^{8} + \frac{18372467}{1398016} a^{7} + \frac{35452205}{1398016} a^{6} - \frac{16496667}{699008} a^{5} + \frac{4517191}{349504} a^{4} + \frac{9277671}{174752} a^{3} - \frac{653149}{43688} a^{2} + \frac{263005}{43688} a + \frac{1000995}{21844} \)
(order $42$)
| 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: |
$\frac{489923}{44736512}a^{23}-\frac{831389}{22368256}a^{22}+\frac{151605}{22368256}a^{21}+\frac{2917971}{22368256}a^{20}-\frac{3919615}{11184128}a^{19}-\frac{1031425}{44736512}a^{18}+\frac{39186535}{44736512}a^{17}-\frac{33189959}{22368256}a^{16}-\frac{32898683}{44736512}a^{15}+\frac{5239301}{1398016}a^{14}-\frac{130082829}{44736512}a^{13}-\frac{236457753}{44736512}a^{12}+\frac{471863437}{44736512}a^{11}+\frac{18022431}{22368256}a^{10}-\frac{15041351}{699008}a^{9}+\frac{26654251}{1398016}a^{8}+\frac{50189805}{2796032}a^{7}-\frac{19068869}{349504}a^{6}+\frac{3359763}{174752}a^{5}+\frac{7241099}{174752}a^{4}-\frac{14665679}{174752}a^{3}+\frac{535345}{87376}a^{2}+\frac{208087}{5461}a-\frac{1441905}{21844}$, $\frac{186711}{11184128}a^{23}-\frac{170287}{5592064}a^{22}-\frac{20233}{5592064}a^{21}+\frac{1024863}{5592064}a^{20}-\frac{760543}{2796032}a^{19}-\frac{1553669}{11184128}a^{18}+\frac{11506911}{11184128}a^{17}-\frac{5702975}{5592064}a^{16}-\frac{14403443}{11184128}a^{15}+\frac{10046929}{2796032}a^{14}-\frac{12853773}{11184128}a^{13}-\frac{71296945}{11184128}a^{12}+\frac{90670289}{11184128}a^{11}+\frac{30163259}{5592064}a^{10}-\frac{56807143}{2796032}a^{9}+\frac{15618205}{1398016}a^{8}+\frac{2289589}{87376}a^{7}-\frac{7456397}{174752}a^{6}+\frac{1182983}{174752}a^{5}+\frac{274461}{5461}a^{4}-\frac{308375}{5461}a^{3}-\frac{25141}{10922}a^{2}+\frac{228026}{5461}a-\frac{215093}{5461}$, $\frac{134005}{44736512}a^{23}-\frac{74595}{2796032}a^{22}+\frac{261975}{22368256}a^{21}+\frac{1084391}{22368256}a^{20}-\frac{716937}{2796032}a^{19}+\frac{2550249}{44736512}a^{18}+\frac{20008239}{44736512}a^{17}-\frac{6398457}{5592064}a^{16}-\frac{6187765}{44736512}a^{15}+\frac{52586189}{22368256}a^{14}-\frac{118750039}{44736512}a^{13}-\frac{118308993}{44736512}a^{12}+\frac{339682453}{44736512}a^{11}-\frac{5144979}{2796032}a^{10}-\frac{76084771}{5592064}a^{9}+\frac{42807823}{2796032}a^{8}+\frac{18256447}{2796032}a^{7}-\frac{53567841}{1398016}a^{6}+\frac{12073355}{699008}a^{5}+\frac{6978059}{349504}a^{4}-\frac{5437783}{87376}a^{3}+\frac{632403}{87376}a^{2}+\frac{879567}{43688}a-\frac{268431}{5461}$, $\frac{730331}{22368256}a^{23}-\frac{898407}{22368256}a^{22}-\frac{145533}{11184128}a^{21}+\frac{486177}{1398016}a^{20}-\frac{3940329}{11184128}a^{19}-\frac{7376225}{22368256}a^{18}+\frac{20614079}{11184128}a^{17}-\frac{26629977}{22368256}a^{16}-\frac{58612991}{22368256}a^{15}+\frac{130991685}{22368256}a^{14}-\frac{5638907}{22368256}a^{13}-\frac{129615177}{11184128}a^{12}+\frac{2928071}{260096}a^{11}+\frac{294326813}{22368256}a^{10}-\frac{182486611}{5592064}a^{9}+\frac{15684997}{1398016}a^{8}+\frac{18296691}{349504}a^{7}-\frac{83364361}{1398016}a^{6}-\frac{395811}{699008}a^{5}+\frac{34182889}{349504}a^{4}-\frac{11789367}{174752}a^{3}-\frac{251985}{21844}a^{2}+\frac{3614327}{43688}a-\frac{905689}{21844}$, $\frac{1244123}{44736512}a^{23}-\frac{40555}{1398016}a^{22}-\frac{381307}{22368256}a^{21}+\frac{6532441}{22368256}a^{20}-\frac{333119}{1398016}a^{19}-\frac{14609161}{44736512}a^{18}+\frac{66264497}{44736512}a^{17}-\frac{1889719}{2796032}a^{16}-\frac{104380883}{44736512}a^{15}+\frac{97469083}{22368256}a^{14}+\frac{36027087}{44736512}a^{13}-\frac{422791535}{44736512}a^{12}+\frac{320543971}{44736512}a^{11}+\frac{70623125}{5592064}a^{10}-\frac{67532219}{2796032}a^{9}+\frac{10725751}{2796032}a^{8}+\frac{122469447}{2796032}a^{7}-\frac{54431957}{1398016}a^{6}-\frac{5834733}{699008}a^{5}+\frac{26806009}{349504}a^{4}-\frac{201999}{5461}a^{3}-\frac{1281385}{87376}a^{2}+\frac{2682535}{43688}a-\frac{202305}{10922}$, $\frac{3273}{699008}a^{23}+\frac{109401}{5592064}a^{22}-\frac{81515}{5592064}a^{21}+\frac{4001}{174752}a^{20}+\frac{6535}{32512}a^{19}-\frac{370899}{2796032}a^{18}-\frac{583183}{5592064}a^{17}+\frac{1389861}{1398016}a^{16}-\frac{2299375}{5592064}a^{15}-\frac{7608131}{5592064}a^{14}+\frac{15899645}{5592064}a^{13}+\frac{2608877}{5592064}a^{12}-\frac{129757}{21844}a^{11}+\frac{13154929}{2796032}a^{10}+\frac{45791091}{5592064}a^{9}-\frac{39453419}{2796032}a^{8}+\frac{1193115}{349504}a^{7}+\frac{20640611}{699008}a^{6}-\frac{1619775}{87376}a^{5}-\frac{181437}{87376}a^{4}+\frac{4627745}{87376}a^{3}-\frac{56667}{5461}a^{2}-\frac{105437}{21844}a+\frac{230309}{5461}$, $\frac{173969}{22368256}a^{23}+\frac{643743}{11184128}a^{22}-\frac{500975}{11184128}a^{21}+\frac{341657}{11184128}a^{20}+\frac{3277683}{5592064}a^{19}-\frac{8326931}{22368256}a^{18}-\frac{9498279}{22368256}a^{17}+\frac{32149455}{11184128}a^{16}-\frac{22707989}{22368256}a^{15}-\frac{23512993}{5592064}a^{14}+\frac{181974085}{22368256}a^{13}+\frac{45361513}{22368256}a^{12}-\frac{396139833}{22368256}a^{11}+\frac{142532357}{11184128}a^{10}+\frac{140011955}{5592064}a^{9}-\frac{117412515}{2796032}a^{8}+\frac{8872591}{1398016}a^{7}+\frac{60839677}{699008}a^{6}-\frac{9660371}{174752}a^{5}-\frac{1252459}{87376}a^{4}+\frac{6777939}{43688}a^{3}-\frac{1320937}{43688}a^{2}-\frac{518917}{21844}a+\frac{1382985}{10922}$, $\frac{349637}{44736512}a^{23}+\frac{99737}{22368256}a^{22}-\frac{93937}{22368256}a^{21}+\frac{1624413}{22368256}a^{20}+\frac{560831}{11184128}a^{19}-\frac{3890631}{44736512}a^{18}+\frac{12620937}{44736512}a^{17}+\frac{6830103}{22368256}a^{16}-\frac{24684389}{44736512}a^{15}+\frac{2116431}{5592064}a^{14}+\frac{58754797}{44736512}a^{13}-\frac{76200935}{44736512}a^{12}-\frac{40602053}{44736512}a^{11}+\frac{99709713}{22368256}a^{10}-\frac{9122525}{5592064}a^{9}-\frac{6482325}{1398016}a^{8}+\frac{31550885}{2796032}a^{7}+\frac{85029}{16256}a^{6}-\frac{2730739}{349504}a^{5}+\frac{3285903}{174752}a^{4}+\frac{78439}{4064}a^{3}-\frac{375391}{87376}a^{2}+\frac{358611}{21844}a+\frac{463835}{21844}$, $\frac{83821}{11184128}a^{23}-\frac{15995}{5592064}a^{22}-\frac{8479}{699008}a^{21}+\frac{439709}{5592064}a^{20}-\frac{9869}{699008}a^{19}-\frac{1675043}{11184128}a^{18}+\frac{4219881}{11184128}a^{17}+\frac{27335}{699008}a^{16}-\frac{8941991}{11184128}a^{15}+\frac{1310763}{1398016}a^{14}+\frac{8849851}{11184128}a^{13}-\frac{28555619}{11184128}a^{12}+\frac{9360573}{11184128}a^{11}+\frac{1541543}{349504}a^{10}-\frac{29120711}{5592064}a^{9}-\frac{2651881}{1398016}a^{8}+\frac{18562581}{1398016}a^{7}-\frac{4716031}{699008}a^{6}-\frac{2417435}{349504}a^{5}+\frac{2038403}{87376}a^{4}-\frac{563199}{87376}a^{3}-\frac{306317}{43688}a^{2}+\frac{437321}{21844}a-\frac{58325}{10922}$, $\frac{1162609}{44736512}a^{23}+\frac{296539}{5592064}a^{22}-\frac{1655905}{22368256}a^{21}+\frac{4422147}{22368256}a^{20}+\frac{1671683}{2796032}a^{19}-\frac{31383867}{44736512}a^{18}+\frac{11822923}{44736512}a^{17}+\frac{18263247}{5592064}a^{16}-\frac{120945945}{44736512}a^{15}-\frac{66019099}{22368256}a^{14}+\frac{484184093}{44736512}a^{13}-\frac{122101493}{44736512}a^{12}-\frac{829338415}{44736512}a^{11}+\frac{64890479}{2796032}a^{10}+\frac{51238963}{2796032}a^{9}-\frac{144115637}{2796032}a^{8}+\frac{87974793}{2796032}a^{7}+\frac{121566545}{1398016}a^{6}-\frac{53327807}{699008}a^{5}+\frac{7745131}{349504}a^{4}+\frac{1847455}{10922}a^{3}-\frac{4271911}{87376}a^{2}+\frac{124593}{43688}a+\frac{1527021}{10922}$, $\frac{129293}{22368256}a^{23}-\frac{266221}{22368256}a^{22}+\frac{29747}{11184128}a^{21}+\frac{41965}{699008}a^{20}-\frac{1267039}{11184128}a^{19}-\frac{458623}{22368256}a^{18}+\frac{3828851}{11184128}a^{17}-\frac{10454455}{22368256}a^{16}-\frac{7813757}{22368256}a^{15}+\frac{29194663}{22368256}a^{14}-\frac{17945473}{22368256}a^{13}-\frac{23276989}{11184128}a^{12}+\frac{38553015}{11184128}a^{11}+\frac{22254843}{22368256}a^{10}-\frac{10375893}{1398016}a^{9}+\frac{17019693}{2796032}a^{8}+\frac{2728109}{349504}a^{7}-\frac{24416857}{1398016}a^{6}+\frac{3998411}{699008}a^{5}+\frac{5506827}{349504}a^{4}-\frac{4624383}{174752}a^{3}+\frac{80}{127}a^{2}+\frac{578143}{43688}a-\frac{456535}{21844}$
| 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: | \( 9137650.497592166 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 9137650.497592166 \cdot 3}{42\cdot\sqrt{138359014736314946502328332753681}}\cr\approx \mathstrut & 0.210068627107942 \end{aligned}\] (assuming GRH)
Galois group
$C_2^3\times A_4$ (as 24T135):
A solvable group of order 96 |
The 32 conjugacy class representatives for $C_2^3\times A_4$ |
Character table for $C_2^3\times A_4$ is not computed |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{4}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{4}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{12}$ | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{16}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\)
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
\(7\)
| 7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |
7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ | |
\(239\)
| 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 $4$ | $2$ | $2$ | $2$ |