Normalized defining polynomial
\( x^{13} - 13x^{8} + 39x^{7} - 78x^{6} + 117x^{5} - 130x^{4} + 104x^{3} - 65x^{2} + 26x - 5 \)
Invariants
| Degree: | $13$ |
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| Signature: | $[1, 6]$ |
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| Discriminant: |
\(51185893014090757\)
\(\medspace = 13^{15}\)
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| Root discriminant: | \(19.29\) |
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| Galois root discriminant: | $13^{63/52}\approx 22.365840080473813$ | ||
| Ramified primes: |
\(13\)
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| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{34}a^{10}-\frac{3}{17}a^{9}+\frac{6}{17}a^{8}-\frac{15}{34}a^{7}+\frac{1}{34}a^{6}+\frac{11}{34}a^{5}-\frac{2}{17}a^{4}-\frac{7}{17}a^{3}+\frac{3}{34}a^{2}+\frac{9}{34}a+\frac{3}{17}$, $\frac{1}{34}a^{11}-\frac{7}{34}a^{9}-\frac{11}{34}a^{8}+\frac{13}{34}a^{7}+\frac{11}{34}a^{5}+\frac{13}{34}a^{4}-\frac{13}{34}a^{3}-\frac{7}{34}a^{2}+\frac{9}{34}a-\frac{15}{34}$, $\frac{1}{578}a^{12}+\frac{3}{289}a^{11}+\frac{1}{289}a^{10}+\frac{131}{578}a^{9}+\frac{21}{578}a^{8}+\frac{181}{578}a^{7}+\frac{27}{289}a^{6}+\frac{38}{289}a^{5}+\frac{63}{578}a^{4}+\frac{197}{578}a^{3}+\frac{31}{289}a^{2}-\frac{127}{289}a+\frac{67}{289}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
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| Narrow class group: | Trivial group, which has order $1$ |
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Unit group
| Rank: | $6$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{157}{578}a^{12}+\frac{80}{289}a^{11}+\frac{21}{289}a^{10}-\frac{71}{578}a^{9}-\frac{35}{578}a^{8}-\frac{1945}{578}a^{7}+\frac{2080}{289}a^{6}-\frac{3299}{289}a^{5}+\frac{8327}{578}a^{4}-\frac{7117}{578}a^{3}+\frac{1994}{289}a^{2}-\frac{984}{289}a+\frac{251}{289}$, $\frac{29}{289}a^{12}+\frac{25}{578}a^{11}+\frac{99}{578}a^{10}+\frac{135}{578}a^{9}-\frac{57}{578}a^{8}-\frac{480}{289}a^{7}+\frac{1959}{578}a^{6}-\frac{2267}{289}a^{5}+\frac{7037}{578}a^{4}-\frac{6679}{578}a^{3}+\frac{2036}{289}a^{2}-\frac{804}{289}a-\frac{201}{578}$, $\frac{141}{578}a^{12}-\frac{53}{289}a^{11}-\frac{109}{578}a^{10}+\frac{13}{289}a^{9}+\frac{71}{578}a^{8}-\frac{899}{289}a^{7}+\frac{3467}{289}a^{6}-\frac{6797}{289}a^{5}+\frac{10162}{289}a^{4}-\frac{21999}{578}a^{3}+\frac{15967}{578}a^{2}-\frac{3865}{289}a+\frac{1639}{578}$, $\frac{195}{289}a^{12}+\frac{351}{578}a^{11}+\frac{118}{289}a^{10}+\frac{73}{578}a^{9}-\frac{157}{578}a^{8}-\frac{5485}{578}a^{7}+\frac{5056}{289}a^{6}-\frac{20187}{578}a^{5}+\frac{26321}{578}a^{4}-\frac{23215}{578}a^{3}+\frac{16241}{578}a^{2}-\frac{7413}{578}a+\frac{1379}{578}$, $\frac{25}{578}a^{12}+\frac{99}{578}a^{11}+\frac{135}{578}a^{10}-\frac{57}{578}a^{9}-\frac{103}{289}a^{8}-\frac{303}{578}a^{7}-\frac{5}{289}a^{6}+\frac{251}{578}a^{5}+\frac{861}{578}a^{4}-\frac{980}{289}a^{3}+\frac{1081}{289}a^{2}-\frac{2287}{578}a+\frac{434}{289}$, $\frac{183}{578}a^{12}-\frac{80}{289}a^{11}-\frac{127}{578}a^{10}+\frac{95}{289}a^{9}+\frac{205}{578}a^{8}-\frac{1212}{289}a^{7}+\frac{4550}{289}a^{6}-\frac{9468}{289}a^{5}+\frac{13168}{289}a^{4}-\frac{27971}{578}a^{3}+\frac{18673}{578}a^{2}-\frac{3232}{289}a+\frac{841}{578}$
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| Regulator: | \( 2655.34526727 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{6}\cdot 2655.34526727 \cdot 1}{2\cdot\sqrt{51185893014090757}}\cr\approx \mathstrut & 0.722146094854 \end{aligned}\]
Galois group
$C_{13}:C_4$ (as 13T4):
| A solvable group of order 52 |
| The 7 conjugacy class representatives for $C_{13}:C_4$ |
| Character table for $C_{13}:C_4$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 26 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 | ${\href{/padicField/2.4.0.1}{4} }^{3}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.13.0.1}{13} }$ | ${\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.13.0.1}{13} }$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.13.0.1}{13} }$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.1.13.15a1.4 | $x^{13} + 117 x^{3} + 13$ | $13$ | $1$ | $15$ | $C_{13}:C_4$ | $$[\frac{5}{4}]_{4}$$ |