Normalized defining polynomial
\( x^{16} - 4 x^{15} - 23 x^{14} + 98 x^{13} + 246 x^{12} - 1492 x^{11} + 537 x^{10} + 7567 x^{9} + \cdots + 58841 \)
Invariants
| Degree: | $16$ |
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| Signature: | $[0, 8]$ |
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| Discriminant: |
\(8243206936713178643875538610721\)
\(\medspace = 13^{12}\cdot 29^{12}\)
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| Root discriminant: | \(85.56\) |
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| Galois root discriminant: | $13^{3/4}29^{3/4}\approx 85.55709155460606$ | ||
| Ramified primes: |
\(13\), \(29\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_4:C_4$ |
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| This field is Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{214}a^{11}+\frac{31}{214}a^{10}-\frac{21}{107}a^{9}+\frac{15}{107}a^{8}-\frac{59}{214}a^{7}+\frac{95}{214}a^{6}-\frac{43}{214}a^{5}-\frac{7}{107}a^{4}+\frac{46}{107}a^{3}+\frac{81}{214}a^{2}-\frac{23}{107}a-\frac{53}{107}$, $\frac{1}{187892}a^{12}-\frac{3}{187892}a^{11}+\frac{25547}{187892}a^{10}+\frac{495}{187892}a^{9}-\frac{5486}{46973}a^{8}-\frac{467}{187892}a^{7}+\frac{35225}{93946}a^{6}-\frac{39961}{187892}a^{5}+\frac{13945}{46973}a^{4}+\frac{40395}{187892}a^{3}-\frac{29015}{187892}a^{2}+\frac{5233}{46973}a+\frac{54001}{187892}$, $\frac{1}{187892}a^{13}+\frac{19}{46973}a^{11}+\frac{19691}{93946}a^{10}+\frac{15539}{187892}a^{9}+\frac{15355}{187892}a^{8}+\frac{68171}{187892}a^{7}+\frac{7203}{187892}a^{6}-\frac{2643}{187892}a^{5}-\frac{93419}{187892}a^{4}-\frac{22394}{46973}a^{3}-\frac{61723}{187892}a^{2}-\frac{27195}{187892}a-\frac{51351}{187892}$, $\frac{1}{187892}a^{14}+\frac{25}{46973}a^{11}+\frac{43321}{187892}a^{10}+\frac{40073}{187892}a^{9}-\frac{13061}{187892}a^{8}-\frac{68811}{187892}a^{7}-\frac{91477}{187892}a^{6}+\frac{39193}{187892}a^{5}-\frac{4469}{46973}a^{4}+\frac{91395}{187892}a^{3}+\frac{11041}{187892}a^{2}+\frac{81331}{187892}a+\frac{20996}{46973}$, $\frac{1}{89\cdots 16}a^{15}-\frac{43\cdots 83}{22\cdots 79}a^{14}+\frac{30\cdots 09}{89\cdots 16}a^{13}+\frac{11\cdots 09}{89\cdots 16}a^{12}+\frac{59\cdots 09}{44\cdots 58}a^{11}-\frac{55\cdots 13}{44\cdots 58}a^{10}+\frac{98\cdots 23}{89\cdots 16}a^{9}-\frac{18\cdots 38}{22\cdots 79}a^{8}+\frac{32\cdots 11}{89\cdots 16}a^{7}+\frac{31\cdots 89}{44\cdots 58}a^{6}-\frac{14\cdots 73}{44\cdots 58}a^{5}+\frac{72\cdots 66}{22\cdots 79}a^{4}-\frac{91\cdots 15}{44\cdots 58}a^{3}-\frac{38\cdots 21}{89\cdots 16}a^{2}+\frac{62\cdots 31}{89\cdots 16}a-\frac{11\cdots 37}{44\cdots 58}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{12}\times C_{12}$, which has order $144$ (assuming GRH) |
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| Narrow class group: | $C_{12}\times C_{12}$, which has order $144$ (assuming GRH) |
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| Relative class number: | $144$ (assuming GRH) |
Unit group
| Rank: | $7$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{17\cdots 97}{44\cdots 58}a^{15}-\frac{17\cdots 91}{89\cdots 16}a^{14}-\frac{17\cdots 46}{22\cdots 79}a^{13}+\frac{46\cdots 85}{89\cdots 16}a^{12}+\frac{55\cdots 65}{89\cdots 16}a^{11}-\frac{35\cdots 51}{44\cdots 58}a^{10}+\frac{17\cdots 60}{20\cdots 97}a^{9}+\frac{37\cdots 63}{89\cdots 16}a^{8}-\frac{51\cdots 51}{44\cdots 58}a^{7}+\frac{83\cdots 95}{89\cdots 16}a^{6}+\frac{98\cdots 84}{22\cdots 79}a^{5}-\frac{23\cdots 37}{44\cdots 58}a^{4}+\frac{19\cdots 23}{22\cdots 79}a^{3}+\frac{20\cdots 10}{22\cdots 79}a^{2}-\frac{62\cdots 61}{89\cdots 16}a+\frac{44\cdots 87}{89\cdots 16}$, $\frac{33\cdots 11}{89\cdots 16}a^{15}-\frac{16\cdots 71}{89\cdots 16}a^{14}-\frac{31\cdots 39}{44\cdots 58}a^{13}+\frac{40\cdots 55}{89\cdots 16}a^{12}+\frac{23\cdots 61}{44\cdots 58}a^{11}-\frac{57\cdots 65}{89\cdots 16}a^{10}+\frac{68\cdots 71}{89\cdots 16}a^{9}+\frac{58\cdots 92}{22\cdots 79}a^{8}-\frac{69\cdots 69}{89\cdots 16}a^{7}+\frac{80\cdots 47}{22\cdots 79}a^{6}+\frac{11\cdots 21}{44\cdots 58}a^{5}-\frac{27\cdots 67}{89\cdots 16}a^{4}+\frac{19\cdots 31}{89\cdots 16}a^{3}+\frac{68\cdots 51}{89\cdots 16}a^{2}+\frac{39\cdots 95}{89\cdots 16}a-\frac{37\cdots 39}{89\cdots 16}$, $\frac{6422282}{74357590621183}a^{15}+\frac{234530571}{74357590621183}a^{14}-\frac{793143192}{74357590621183}a^{13}-\frac{6932269894}{74357590621183}a^{12}+\frac{17403713606}{74357590621183}a^{11}+\frac{94643823731}{74357590621183}a^{10}-\frac{272762732447}{74357590621183}a^{9}-\frac{442579958616}{74357590621183}a^{8}+\frac{1914533514433}{74357590621183}a^{7}+\frac{321746111967}{74357590621183}a^{6}-\frac{5053762395849}{74357590621183}a^{5}+\frac{9911102955808}{74357590621183}a^{4}+\frac{25171363320457}{74357590621183}a^{3}+\frac{19656243400414}{74357590621183}a^{2}+\frac{7337632379887}{74357590621183}a+\frac{27121527550912}{74357590621183}$, $\frac{469059064393}{22\cdots 36}a^{15}-\frac{2218221216177}{22\cdots 36}a^{14}-\frac{9545582523355}{22\cdots 36}a^{13}+\frac{54871547166995}{22\cdots 36}a^{12}+\frac{20549929920097}{56\cdots 59}a^{11}-\frac{807457451980453}{22\cdots 36}a^{10}+\frac{197905646344745}{56\cdots 59}a^{9}+\frac{36\cdots 61}{22\cdots 36}a^{8}-\frac{24\cdots 69}{56\cdots 59}a^{7}+\frac{32\cdots 47}{22\cdots 36}a^{6}+\frac{34\cdots 15}{22\cdots 36}a^{5}-\frac{10\cdots 55}{56\cdots 59}a^{4}+\frac{33\cdots 93}{22\cdots 36}a^{3}+\frac{26\cdots 45}{56\cdots 59}a^{2}+\frac{16\cdots 88}{56\cdots 59}a+\frac{65210894162613}{53262722236837}$, $\frac{43\cdots 83}{22\cdots 79}a^{15}-\frac{73\cdots 99}{89\cdots 16}a^{14}-\frac{43\cdots 27}{89\cdots 16}a^{13}+\frac{20\cdots 43}{89\cdots 16}a^{12}+\frac{45\cdots 81}{89\cdots 16}a^{11}-\frac{16\cdots 29}{44\cdots 58}a^{10}+\frac{13\cdots 99}{89\cdots 16}a^{9}+\frac{52\cdots 60}{22\cdots 79}a^{8}-\frac{43\cdots 23}{89\cdots 16}a^{7}-\frac{38\cdots 39}{44\cdots 58}a^{6}+\frac{19\cdots 43}{89\cdots 16}a^{5}-\frac{22\cdots 13}{89\cdots 16}a^{4}+\frac{49\cdots 49}{44\cdots 58}a^{3}+\frac{47\cdots 95}{89\cdots 16}a^{2}+\frac{33\cdots 59}{22\cdots 79}a-\frac{24\cdots 29}{44\cdots 58}$, $\frac{35\cdots 95}{44\cdots 58}a^{15}-\frac{37\cdots 63}{89\cdots 16}a^{14}-\frac{20\cdots 53}{89\cdots 16}a^{13}+\frac{10\cdots 67}{89\cdots 16}a^{12}+\frac{32\cdots 27}{89\cdots 16}a^{11}-\frac{38\cdots 81}{22\cdots 79}a^{10}-\frac{17\cdots 05}{89\cdots 16}a^{9}+\frac{23\cdots 25}{22\cdots 79}a^{8}+\frac{47\cdots 29}{89\cdots 16}a^{7}-\frac{86\cdots 51}{44\cdots 58}a^{6}-\frac{40\cdots 27}{89\cdots 16}a^{5}-\frac{63\cdots 21}{89\cdots 16}a^{4}-\frac{34\cdots 65}{22\cdots 79}a^{3}-\frac{11\cdots 05}{89\cdots 16}a^{2}-\frac{18\cdots 99}{44\cdots 58}a-\frac{52\cdots 19}{22\cdots 79}$, $\frac{42\cdots 61}{44\cdots 58}a^{15}-\frac{12\cdots 42}{22\cdots 79}a^{14}-\frac{45\cdots 91}{44\cdots 58}a^{13}+\frac{25\cdots 64}{22\cdots 79}a^{12}-\frac{11\cdots 33}{44\cdots 58}a^{11}-\frac{63\cdots 43}{44\cdots 58}a^{10}+\frac{76\cdots 49}{22\cdots 79}a^{9}+\frac{25\cdots 15}{22\cdots 79}a^{8}-\frac{29\cdots 36}{22\cdots 79}a^{7}+\frac{60\cdots 34}{22\cdots 79}a^{6}-\frac{11\cdots 97}{44\cdots 58}a^{5}-\frac{25\cdots 42}{22\cdots 79}a^{4}+\frac{28\cdots 71}{44\cdots 58}a^{3}+\frac{22\cdots 97}{22\cdots 79}a^{2}+\frac{78\cdots 15}{44\cdots 58}a+\frac{12\cdots 79}{44\cdots 58}$
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| Regulator: | \( 2985208.11128 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2985208.11128 \cdot 144}{2\cdot\sqrt{8243206936713178643875538610721}}\cr\approx \mathstrut & 0.181843572544 \end{aligned}\] (assuming GRH)
Galois group
| A solvable group of order 16 |
| The 10 conjugacy class representatives for $C_4:C_4$ |
| Character table for $C_4:C_4$ |
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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.2.4.6a1.2 | $x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24216 x^{4} + 14400 x^{3} + 3488 x^{2} + 384 x + 29$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ |
| 13.2.4.6a1.2 | $x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24216 x^{4} + 14400 x^{3} + 3488 x^{2} + 384 x + 29$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ | |
|
\(29\)
| 29.1.4.3a1.3 | $x^{4} + 116$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 29.1.4.3a1.3 | $x^{4} + 116$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 29.1.4.3a1.3 | $x^{4} + 116$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 29.1.4.3a1.3 | $x^{4} + 116$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |