Normalized defining polynomial
\( x^{44} + 63 x^{42} + 1771 x^{40} + 29412 x^{38} + 322363 x^{36} + 2470134 x^{34} + 13694893 x^{32} + \cdots + 1 \)
Invariants
Degree: | $44$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 22]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(123\!\cdots\!000\) \(\medspace = 2^{44}\cdot 5^{22}\cdot 23^{40}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(77.35\) | 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: | $2\cdot 5^{1/2}23^{10/11}\approx 77.34805443730704$ | ||
Ramified primes: | \(2\), \(5\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $44$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(460=2^{2}\cdot 5\cdot 23\) | ||
Dirichlet character group: | $\lbrace$$\chi_{460}(1,·)$, $\chi_{460}(259,·)$, $\chi_{460}(261,·)$, $\chi_{460}(9,·)$, $\chi_{460}(139,·)$, $\chi_{460}(269,·)$, $\chi_{460}(271,·)$, $\chi_{460}(131,·)$, $\chi_{460}(279,·)$, $\chi_{460}(409,·)$, $\chi_{460}(29,·)$, $\chi_{460}(31,·)$, $\chi_{460}(289,·)$, $\chi_{460}(39,·)$, $\chi_{460}(41,·)$, $\chi_{460}(301,·)$, $\chi_{460}(49,·)$, $\chi_{460}(179,·)$, $\chi_{460}(439,·)$, $\chi_{460}(441,·)$, $\chi_{460}(59,·)$, $\chi_{460}(151,·)$, $\chi_{460}(449,·)$, $\chi_{460}(331,·)$, $\chi_{460}(71,·)$, $\chi_{460}(311,·)$, $\chi_{460}(141,·)$, $\chi_{460}(81,·)$, $\chi_{460}(211,·)$, $\chi_{460}(399,·)$, $\chi_{460}(219,·)$, $\chi_{460}(349,·)$, $\chi_{460}(351,·)$, $\chi_{460}(101,·)$, $\chi_{460}(209,·)$, $\chi_{460}(361,·)$, $\chi_{460}(231,·)$, $\chi_{460}(239,·)$, $\chi_{460}(369,·)$, $\chi_{460}(371,·)$, $\chi_{460}(169,·)$, $\chi_{460}(121,·)$, $\chi_{460}(119,·)$, $\chi_{460}(381,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{2097152}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $\frac{1}{139}a^{40}+\frac{45}{139}a^{38}+\frac{28}{139}a^{36}-\frac{11}{139}a^{34}-\frac{50}{139}a^{32}+\frac{8}{139}a^{30}-\frac{2}{139}a^{28}+\frac{12}{139}a^{26}+\frac{64}{139}a^{24}+\frac{19}{139}a^{22}-\frac{44}{139}a^{20}+\frac{32}{139}a^{18}+\frac{3}{139}a^{16}-\frac{67}{139}a^{14}+\frac{53}{139}a^{12}+\frac{65}{139}a^{10}-\frac{57}{139}a^{8}-\frac{3}{139}a^{6}-\frac{11}{139}a^{4}-\frac{10}{139}a^{2}+\frac{66}{139}$, $\frac{1}{139}a^{41}+\frac{45}{139}a^{39}+\frac{28}{139}a^{37}-\frac{11}{139}a^{35}-\frac{50}{139}a^{33}+\frac{8}{139}a^{31}-\frac{2}{139}a^{29}+\frac{12}{139}a^{27}+\frac{64}{139}a^{25}+\frac{19}{139}a^{23}-\frac{44}{139}a^{21}+\frac{32}{139}a^{19}+\frac{3}{139}a^{17}-\frac{67}{139}a^{15}+\frac{53}{139}a^{13}+\frac{65}{139}a^{11}-\frac{57}{139}a^{9}-\frac{3}{139}a^{7}-\frac{11}{139}a^{5}-\frac{10}{139}a^{3}+\frac{66}{139}a$, $\frac{1}{90\!\cdots\!99}a^{42}+\frac{13\!\cdots\!30}{90\!\cdots\!99}a^{40}+\frac{99\!\cdots\!52}{90\!\cdots\!99}a^{38}+\frac{33\!\cdots\!27}{90\!\cdots\!99}a^{36}-\frac{73\!\cdots\!60}{90\!\cdots\!99}a^{34}+\frac{32\!\cdots\!15}{90\!\cdots\!99}a^{32}+\frac{43\!\cdots\!17}{90\!\cdots\!99}a^{30}+\frac{64\!\cdots\!51}{90\!\cdots\!99}a^{28}-\frac{13\!\cdots\!46}{90\!\cdots\!99}a^{26}+\frac{25\!\cdots\!82}{90\!\cdots\!99}a^{24}-\frac{17\!\cdots\!53}{90\!\cdots\!99}a^{22}+\frac{34\!\cdots\!96}{90\!\cdots\!99}a^{20}+\frac{13\!\cdots\!00}{90\!\cdots\!99}a^{18}+\frac{33\!\cdots\!89}{90\!\cdots\!99}a^{16}+\frac{28\!\cdots\!98}{90\!\cdots\!99}a^{14}+\frac{23\!\cdots\!31}{90\!\cdots\!99}a^{12}+\frac{14\!\cdots\!94}{90\!\cdots\!99}a^{10}-\frac{23\!\cdots\!01}{90\!\cdots\!99}a^{8}+\frac{17\!\cdots\!00}{90\!\cdots\!99}a^{6}-\frac{18\!\cdots\!35}{90\!\cdots\!99}a^{4}+\frac{72\!\cdots\!99}{90\!\cdots\!99}a^{2}-\frac{47\!\cdots\!11}{90\!\cdots\!99}$, $\frac{1}{90\!\cdots\!99}a^{43}+\frac{13\!\cdots\!30}{90\!\cdots\!99}a^{41}+\frac{99\!\cdots\!52}{90\!\cdots\!99}a^{39}+\frac{33\!\cdots\!27}{90\!\cdots\!99}a^{37}-\frac{73\!\cdots\!60}{90\!\cdots\!99}a^{35}+\frac{32\!\cdots\!15}{90\!\cdots\!99}a^{33}+\frac{43\!\cdots\!17}{90\!\cdots\!99}a^{31}+\frac{64\!\cdots\!51}{90\!\cdots\!99}a^{29}-\frac{13\!\cdots\!46}{90\!\cdots\!99}a^{27}+\frac{25\!\cdots\!82}{90\!\cdots\!99}a^{25}-\frac{17\!\cdots\!53}{90\!\cdots\!99}a^{23}+\frac{34\!\cdots\!96}{90\!\cdots\!99}a^{21}+\frac{13\!\cdots\!00}{90\!\cdots\!99}a^{19}+\frac{33\!\cdots\!89}{90\!\cdots\!99}a^{17}+\frac{28\!\cdots\!98}{90\!\cdots\!99}a^{15}+\frac{23\!\cdots\!31}{90\!\cdots\!99}a^{13}+\frac{14\!\cdots\!94}{90\!\cdots\!99}a^{11}-\frac{23\!\cdots\!01}{90\!\cdots\!99}a^{9}+\frac{17\!\cdots\!00}{90\!\cdots\!99}a^{7}-\frac{18\!\cdots\!35}{90\!\cdots\!99}a^{5}+\frac{72\!\cdots\!99}{90\!\cdots\!99}a^{3}-\frac{47\!\cdots\!11}{90\!\cdots\!99}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $21$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{209157351019064485193720}{2226090219617027257916699} a^{43} + \frac{12995488274359730586935290}{2226090219617027257916699} a^{41} + \frac{359143284978099066457720105}{2226090219617027257916699} a^{39} + \frac{5839972721540257020635425985}{2226090219617027257916699} a^{37} + \frac{62348418235602803927625805125}{2226090219617027257916699} a^{35} + \frac{462315626899471636311969926105}{2226090219617027257916699} a^{33} + \frac{2459676882912455947235289455830}{2226090219617027257916699} a^{31} + \frac{9577529956515647388992261040325}{2226090219617027257916699} a^{29} + \frac{27616587656070514976128092203875}{2226090219617027257916699} a^{27} + \frac{59278266630197454362918907321225}{2226090219617027257916699} a^{25} + \frac{94532141338580093200503304010726}{2226090219617027257916699} a^{23} + \frac{110623091364929624646755757921827}{2226090219617027257916699} a^{21} + \frac{92010792085643524841366328007535}{2226090219617027257916699} a^{19} + \frac{50081888735077478032214559584484}{2226090219617027257916699} a^{17} + \frac{12880295636412651288925218674382}{2226090219617027257916699} a^{15} - \frac{3673808100033226154684241754493}{2226090219617027257916699} a^{13} - \frac{4851804509885062421451654632776}{2226090219617027257916699} a^{11} - \frac{2053398889912630968727299729775}{2226090219617027257916699} a^{9} - \frac{460153217780433676988443046173}{2226090219617027257916699} a^{7} - \frac{54333828788409951784086810349}{2226090219617027257916699} a^{5} - \frac{2882529697730366774413556719}{2226090219617027257916699} a^{3} - \frac{44310276008005938920339643}{2226090219617027257916699} a \) (order $4$) | 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: | not computed | 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: | not computed | 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 $
Galois group
$C_2\times C_{22}$ (as 44T2):
An abelian group of order 44 |
The 44 conjugacy class representatives for $C_2\times C_{22}$ |
Character table for $C_2\times C_{22}$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | $22^{2}$ | R | $22^{2}$ | $22^{2}$ | $22^{2}$ | $22^{2}$ | $22^{2}$ | R | ${\href{/padicField/29.11.0.1}{11} }^{4}$ | $22^{2}$ | $22^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{4}$ | $22^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{22}$ | $22^{2}$ | $22^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $44$ | $2$ | $22$ | $44$ | |||
\(5\) | 5.22.11.1 | $x^{22} + 220 x^{21} + 22055 x^{20} + 1331000 x^{19} + 53791375 x^{18} + 1531447500 x^{17} + 31435820625 x^{16} + 467679300000 x^{15} + 4991151206250 x^{14} + 37171668875000 x^{13} + 183624733943756 x^{12} + 553513923250726 x^{11} + 918123669784090 x^{10} + 929291725767350 x^{9} + 623894056087500 x^{8} + 292303912609500 x^{7} + 98324330218125 x^{6} + 25190924781000 x^{5} + 17099014728125 x^{4} + 90189081743750 x^{3} + 391939091809384 x^{2} + 906877245981448 x + 669277565422109$ | $2$ | $11$ | $11$ | 22T1 | $[\ ]_{2}^{11}$ |
5.22.11.1 | $x^{22} + 220 x^{21} + 22055 x^{20} + 1331000 x^{19} + 53791375 x^{18} + 1531447500 x^{17} + 31435820625 x^{16} + 467679300000 x^{15} + 4991151206250 x^{14} + 37171668875000 x^{13} + 183624733943756 x^{12} + 553513923250726 x^{11} + 918123669784090 x^{10} + 929291725767350 x^{9} + 623894056087500 x^{8} + 292303912609500 x^{7} + 98324330218125 x^{6} + 25190924781000 x^{5} + 17099014728125 x^{4} + 90189081743750 x^{3} + 391939091809384 x^{2} + 906877245981448 x + 669277565422109$ | $2$ | $11$ | $11$ | 22T1 | $[\ ]_{2}^{11}$ | |
\(23\) | 23.22.20.1 | $x^{22} + 231 x^{21} + 24310 x^{20} + 1539615 x^{19} + 65271580 x^{18} + 1948237137 x^{17} + 41892053577 x^{16} + 652037968890 x^{15} + 7265319209115 x^{14} + 56290884204555 x^{13} + 287278948122936 x^{12} + 881069580352567 x^{11} + 1436394740619993 x^{10} + 1407272105659090 x^{9} + 908164935089445 x^{8} + 407525140102740 x^{7} + 130953632793951 x^{6} + 31291611122541 x^{5} + 17709125163290 x^{4} + 131485551467820 x^{3} + 905728303498040 x^{2} + 3760233666129578 x + 7096289029978197$ | $11$ | $2$ | $20$ | 22T1 | $[\ ]_{11}^{2}$ |
23.22.20.1 | $x^{22} + 231 x^{21} + 24310 x^{20} + 1539615 x^{19} + 65271580 x^{18} + 1948237137 x^{17} + 41892053577 x^{16} + 652037968890 x^{15} + 7265319209115 x^{14} + 56290884204555 x^{13} + 287278948122936 x^{12} + 881069580352567 x^{11} + 1436394740619993 x^{10} + 1407272105659090 x^{9} + 908164935089445 x^{8} + 407525140102740 x^{7} + 130953632793951 x^{6} + 31291611122541 x^{5} + 17709125163290 x^{4} + 131485551467820 x^{3} + 905728303498040 x^{2} + 3760233666129578 x + 7096289029978197$ | $11$ | $2$ | $20$ | 22T1 | $[\ ]_{11}^{2}$ |