Normalized defining polynomial
\( x^{44} - x^{43} + 5 x^{42} + 233 x^{41} - 43 x^{40} - 317 x^{39} + 21354 x^{38} - 8471 x^{37} + \cdots + 10\!\cdots\!29 \)
Invariants
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $\frac{1}{79}a^{37}-\frac{38}{79}a^{36}-\frac{37}{79}a^{35}+\frac{32}{79}a^{34}+\frac{39}{79}a^{33}+\frac{1}{79}a^{32}-\frac{11}{79}a^{31}+\frac{2}{79}a^{30}+\frac{19}{79}a^{29}-\frac{15}{79}a^{28}+\frac{6}{79}a^{27}+\frac{8}{79}a^{26}-\frac{11}{79}a^{25}-\frac{24}{79}a^{24}-\frac{14}{79}a^{23}+\frac{38}{79}a^{22}+\frac{12}{79}a^{21}+\frac{31}{79}a^{20}-\frac{15}{79}a^{19}+\frac{9}{79}a^{18}+\frac{33}{79}a^{17}-\frac{2}{79}a^{16}+\frac{11}{79}a^{15}+\frac{29}{79}a^{14}+\frac{35}{79}a^{13}+\frac{33}{79}a^{12}-\frac{13}{79}a^{11}+\frac{26}{79}a^{10}+\frac{18}{79}a^{9}+\frac{9}{79}a^{8}+\frac{26}{79}a^{7}-\frac{8}{79}a^{6}+\frac{13}{79}a^{5}-\frac{30}{79}a^{4}+\frac{3}{79}a^{3}+\frac{7}{79}a^{2}-\frac{7}{79}a$, $\frac{1}{79}a^{38}+\frac{20}{79}a^{36}-\frac{31}{79}a^{35}-\frac{9}{79}a^{34}-\frac{18}{79}a^{33}+\frac{27}{79}a^{32}-\frac{21}{79}a^{31}+\frac{16}{79}a^{30}-\frac{4}{79}a^{29}-\frac{11}{79}a^{28}-\frac{1}{79}a^{27}-\frac{23}{79}a^{26}+\frac{32}{79}a^{25}+\frac{22}{79}a^{24}-\frac{20}{79}a^{23}+\frac{34}{79}a^{22}+\frac{13}{79}a^{21}-\frac{22}{79}a^{20}-\frac{8}{79}a^{19}-\frac{20}{79}a^{18}-\frac{12}{79}a^{17}+\frac{14}{79}a^{16}-\frac{27}{79}a^{15}+\frac{31}{79}a^{14}+\frac{20}{79}a^{13}-\frac{23}{79}a^{12}+\frac{6}{79}a^{11}-\frac{21}{79}a^{10}-\frac{18}{79}a^{9}-\frac{27}{79}a^{8}+\frac{32}{79}a^{7}+\frac{25}{79}a^{6}-\frac{10}{79}a^{5}-\frac{31}{79}a^{4}-\frac{37}{79}a^{3}+\frac{22}{79}a^{2}-\frac{29}{79}a$, $\frac{1}{79}a^{39}+\frac{18}{79}a^{36}+\frac{20}{79}a^{35}-\frac{26}{79}a^{34}+\frac{37}{79}a^{33}+\frac{38}{79}a^{32}-\frac{1}{79}a^{31}+\frac{35}{79}a^{30}+\frac{4}{79}a^{29}-\frac{17}{79}a^{28}+\frac{15}{79}a^{27}+\frac{30}{79}a^{26}+\frac{5}{79}a^{25}-\frac{14}{79}a^{24}-\frac{2}{79}a^{23}-\frac{36}{79}a^{22}-\frac{25}{79}a^{21}+\frac{4}{79}a^{20}-\frac{36}{79}a^{19}-\frac{34}{79}a^{18}-\frac{14}{79}a^{17}+\frac{13}{79}a^{16}-\frac{31}{79}a^{15}-\frac{7}{79}a^{14}-\frac{12}{79}a^{13}-\frac{22}{79}a^{12}+\frac{2}{79}a^{11}+\frac{15}{79}a^{10}+\frac{8}{79}a^{9}+\frac{10}{79}a^{8}-\frac{21}{79}a^{7}-\frac{8}{79}a^{6}+\frac{25}{79}a^{5}+\frac{10}{79}a^{4}-\frac{38}{79}a^{3}-\frac{11}{79}a^{2}-\frac{18}{79}a$, $\frac{1}{34049}a^{40}+\frac{185}{34049}a^{39}-\frac{148}{34049}a^{38}+\frac{11}{34049}a^{37}-\frac{9772}{34049}a^{36}-\frac{5146}{34049}a^{35}+\frac{14979}{34049}a^{34}+\frac{14962}{34049}a^{33}-\frac{12932}{34049}a^{32}-\frac{10790}{34049}a^{31}-\frac{6252}{34049}a^{30}-\frac{6323}{34049}a^{29}-\frac{15775}{34049}a^{28}+\frac{5202}{34049}a^{27}-\frac{5870}{34049}a^{26}+\frac{7233}{34049}a^{25}+\frac{11700}{34049}a^{24}+\frac{9288}{34049}a^{23}-\frac{11430}{34049}a^{22}+\frac{1982}{34049}a^{21}+\frac{8799}{34049}a^{20}-\frac{6985}{34049}a^{19}+\frac{2044}{34049}a^{18}-\frac{8695}{34049}a^{17}+\frac{133}{431}a^{16}-\frac{14858}{34049}a^{15}-\frac{2069}{34049}a^{14}+\frac{3717}{34049}a^{13}+\frac{13246}{34049}a^{12}+\frac{10806}{34049}a^{11}-\frac{5193}{34049}a^{10}-\frac{15090}{34049}a^{9}+\frac{864}{34049}a^{8}-\frac{12445}{34049}a^{7}-\frac{10787}{34049}a^{6}+\frac{14556}{34049}a^{5}+\frac{14352}{34049}a^{4}-\frac{3482}{34049}a^{3}+\frac{10916}{34049}a^{2}+\frac{16179}{34049}a-\frac{212}{431}$, $\frac{1}{24140741}a^{41}+\frac{74}{24140741}a^{40}+\frac{150855}{24140741}a^{39}+\frac{87123}{24140741}a^{38}-\frac{107106}{24140741}a^{37}+\frac{53088}{305579}a^{36}-\frac{10621108}{24140741}a^{35}+\frac{2097683}{24140741}a^{34}+\frac{3295716}{24140741}a^{33}-\frac{3761992}{24140741}a^{32}-\frac{6251070}{24140741}a^{31}-\frac{7828049}{24140741}a^{30}+\frac{5732657}{24140741}a^{29}-\frac{7611989}{24140741}a^{28}+\frac{11489018}{24140741}a^{27}-\frac{5033845}{24140741}a^{26}-\frac{1245006}{24140741}a^{25}-\frac{6741131}{24140741}a^{24}+\frac{4368807}{24140741}a^{23}-\frac{11124848}{24140741}a^{22}+\frac{4595309}{24140741}a^{21}+\frac{6335999}{24140741}a^{20}-\frac{7722372}{24140741}a^{19}-\frac{1877258}{24140741}a^{18}+\frac{4564589}{24140741}a^{17}+\frac{3184033}{24140741}a^{16}-\frac{10994492}{24140741}a^{15}-\frac{2993521}{24140741}a^{14}-\frac{10026157}{24140741}a^{13}-\frac{10745826}{24140741}a^{12}+\frac{846039}{24140741}a^{11}-\frac{7658699}{24140741}a^{10}-\frac{7581164}{24140741}a^{9}-\frac{6920735}{24140741}a^{8}+\frac{10231537}{24140741}a^{7}-\frac{3378668}{24140741}a^{6}+\frac{1411757}{24140741}a^{5}+\frac{2479413}{24140741}a^{4}+\frac{5820691}{24140741}a^{3}-\frac{2336354}{24140741}a^{2}-\frac{1816927}{24140741}a-\frac{63530}{305579}$, $\frac{1}{15\!\cdots\!89}a^{42}+\frac{295452920}{15\!\cdots\!89}a^{41}+\frac{160035561696}{15\!\cdots\!89}a^{40}-\frac{6699529655516}{15\!\cdots\!89}a^{39}-\frac{13102175018345}{15\!\cdots\!89}a^{38}-\frac{65669465207160}{15\!\cdots\!89}a^{37}-\frac{53\!\cdots\!64}{15\!\cdots\!89}a^{36}-\frac{51\!\cdots\!72}{15\!\cdots\!89}a^{35}+\frac{70\!\cdots\!38}{15\!\cdots\!89}a^{34}-\frac{33\!\cdots\!40}{15\!\cdots\!89}a^{33}+\frac{71\!\cdots\!47}{15\!\cdots\!89}a^{32}+\frac{32\!\cdots\!18}{15\!\cdots\!89}a^{31}-\frac{984508542249974}{15\!\cdots\!89}a^{30}-\frac{57\!\cdots\!22}{15\!\cdots\!89}a^{29}-\frac{67\!\cdots\!35}{15\!\cdots\!89}a^{28}+\frac{41\!\cdots\!46}{15\!\cdots\!89}a^{27}-\frac{44\!\cdots\!81}{15\!\cdots\!89}a^{26}-\frac{46\!\cdots\!89}{15\!\cdots\!89}a^{25}-\frac{71\!\cdots\!73}{15\!\cdots\!89}a^{24}-\frac{34\!\cdots\!47}{15\!\cdots\!89}a^{23}-\frac{474499744952191}{15\!\cdots\!89}a^{22}+\frac{353858735782790}{15\!\cdots\!89}a^{21}-\frac{67\!\cdots\!96}{15\!\cdots\!89}a^{20}-\frac{61\!\cdots\!83}{15\!\cdots\!89}a^{19}+\frac{15\!\cdots\!15}{15\!\cdots\!89}a^{18}-\frac{250745454116064}{15\!\cdots\!89}a^{17}-\frac{59\!\cdots\!99}{15\!\cdots\!89}a^{16}+\frac{42\!\cdots\!81}{15\!\cdots\!89}a^{15}+\frac{49\!\cdots\!76}{15\!\cdots\!89}a^{14}+\frac{49\!\cdots\!59}{15\!\cdots\!89}a^{13}-\frac{39\!\cdots\!54}{15\!\cdots\!89}a^{12}-\frac{17\!\cdots\!22}{15\!\cdots\!89}a^{11}-\frac{34\!\cdots\!10}{15\!\cdots\!89}a^{10}-\frac{12\!\cdots\!81}{15\!\cdots\!89}a^{9}+\frac{77\!\cdots\!95}{15\!\cdots\!89}a^{8}-\frac{60\!\cdots\!95}{15\!\cdots\!89}a^{7}+\frac{24\!\cdots\!03}{15\!\cdots\!89}a^{6}+\frac{19\!\cdots\!46}{15\!\cdots\!89}a^{5}-\frac{59\!\cdots\!71}{15\!\cdots\!89}a^{4}+\frac{23\!\cdots\!82}{15\!\cdots\!89}a^{3}+\frac{69\!\cdots\!64}{15\!\cdots\!89}a^{2}-\frac{869837932516423}{15\!\cdots\!89}a+\frac{21966161071867}{197273010586591}$, $\frac{1}{37\!\cdots\!39}a^{43}-\frac{39\!\cdots\!30}{37\!\cdots\!39}a^{42}-\frac{49\!\cdots\!49}{37\!\cdots\!39}a^{41}-\frac{21\!\cdots\!92}{37\!\cdots\!39}a^{40}+\frac{23\!\cdots\!18}{37\!\cdots\!39}a^{39}-\frac{19\!\cdots\!75}{37\!\cdots\!39}a^{38}-\frac{56\!\cdots\!96}{37\!\cdots\!39}a^{37}+\frac{13\!\cdots\!65}{37\!\cdots\!39}a^{36}-\frac{27\!\cdots\!10}{37\!\cdots\!39}a^{35}+\frac{17\!\cdots\!82}{37\!\cdots\!39}a^{34}+\frac{12\!\cdots\!95}{37\!\cdots\!39}a^{33}-\frac{53\!\cdots\!64}{37\!\cdots\!39}a^{32}-\frac{34\!\cdots\!31}{37\!\cdots\!39}a^{31}+\frac{50\!\cdots\!60}{37\!\cdots\!39}a^{30}-\frac{13\!\cdots\!81}{37\!\cdots\!39}a^{29}-\frac{18\!\cdots\!39}{37\!\cdots\!39}a^{28}+\frac{45\!\cdots\!99}{37\!\cdots\!39}a^{27}+\frac{11\!\cdots\!92}{37\!\cdots\!39}a^{26}+\frac{12\!\cdots\!77}{37\!\cdots\!39}a^{25}+\frac{61\!\cdots\!82}{37\!\cdots\!39}a^{24}+\frac{89\!\cdots\!02}{37\!\cdots\!39}a^{23}+\frac{15\!\cdots\!71}{37\!\cdots\!39}a^{22}-\frac{16\!\cdots\!16}{37\!\cdots\!39}a^{21}-\frac{16\!\cdots\!03}{37\!\cdots\!39}a^{20}+\frac{42\!\cdots\!76}{37\!\cdots\!39}a^{19}+\frac{30\!\cdots\!44}{37\!\cdots\!39}a^{18}-\frac{13\!\cdots\!16}{37\!\cdots\!39}a^{17}+\frac{16\!\cdots\!15}{37\!\cdots\!39}a^{16}+\frac{16\!\cdots\!56}{37\!\cdots\!39}a^{15}-\frac{10\!\cdots\!26}{37\!\cdots\!39}a^{14}-\frac{14\!\cdots\!40}{37\!\cdots\!39}a^{13}+\frac{66\!\cdots\!96}{37\!\cdots\!39}a^{12}-\frac{54\!\cdots\!15}{37\!\cdots\!39}a^{11}+\frac{17\!\cdots\!68}{37\!\cdots\!39}a^{10}-\frac{36\!\cdots\!23}{37\!\cdots\!39}a^{9}+\frac{10\!\cdots\!21}{37\!\cdots\!39}a^{8}+\frac{36\!\cdots\!65}{37\!\cdots\!39}a^{7}-\frac{46\!\cdots\!15}{37\!\cdots\!39}a^{6}-\frac{15\!\cdots\!57}{37\!\cdots\!39}a^{5}-\frac{15\!\cdots\!51}{37\!\cdots\!39}a^{4}-\frac{16\!\cdots\!00}{37\!\cdots\!39}a^{3}-\frac{28\!\cdots\!95}{37\!\cdots\!39}a^{2}-\frac{14\!\cdots\!63}{37\!\cdots\!39}a-\frac{79\!\cdots\!78}{47\!\cdots\!41}$
| 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: |
\( -1 \)
(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: | 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
| A cyclic group of order 44 |
| The 44 conjugacy class representatives for $C_{44}$ |
| Character table for $C_{44}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{397}) \), 4.0.62570773.1, 11.11.97253461433805715000527049.1, 22.22.3754919597060124016902809789203115664418428068917415197.1 |
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 | $44$ | $22^{2}$ | $44$ | $44$ | ${\href{/padicField/11.11.0.1}{11} }^{4}$ | $44$ | $44$ | ${\href{/padicField/19.11.0.1}{11} }^{4}$ | $22^{2}$ | ${\href{/padicField/29.11.0.1}{11} }^{4}$ | ${\href{/padicField/31.11.0.1}{11} }^{4}$ | ${\href{/padicField/37.11.0.1}{11} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{11}$ | $22^{2}$ | ${\href{/padicField/47.11.0.1}{11} }^{4}$ | $44$ | $44$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(397\)
| Deg $44$ | $44$ | $1$ | $43$ |