Normalized defining polynomial
\( x^{26} - 2 x^{25} + 17 x^{24} - 98 x^{23} + 152 x^{22} - 696 x^{21} + 1597 x^{20} + 1896 x^{19} + \cdots + 29070259637 \)
Invariants
| Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-4459331436230036418749915076130198106842162927\)
\(\medspace = -\,17^{13}\cdot 191^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(56.98\) | 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: | $17^{1/2}191^{1/2}\approx 56.9824534396335$ | ||
| Ramified primes: |
\(17\), \(191\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3247}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}+\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{2}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{15}-\frac{1}{2}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{16}-\frac{1}{2}a^{12}-\frac{1}{4}a^{11}+\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{28}a^{17}-\frac{1}{28}a^{16}+\frac{1}{14}a^{15}+\frac{1}{28}a^{14}+\frac{3}{14}a^{12}-\frac{11}{28}a^{11}-\frac{1}{4}a^{10}-\frac{1}{28}a^{9}+\frac{5}{28}a^{8}-\frac{2}{7}a^{7}+\frac{3}{14}a^{6}+\frac{5}{28}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{14}a^{2}-\frac{9}{28}a-\frac{1}{28}$, $\frac{1}{28}a^{18}+\frac{1}{28}a^{16}+\frac{3}{28}a^{15}+\frac{1}{28}a^{14}-\frac{1}{28}a^{13}+\frac{1}{14}a^{12}+\frac{5}{14}a^{11}-\frac{1}{28}a^{10}-\frac{3}{28}a^{9}+\frac{1}{7}a^{8}+\frac{5}{28}a^{7}-\frac{3}{28}a^{6}+\frac{3}{7}a^{5}+\frac{3}{7}a^{3}-\frac{11}{28}a^{2}+\frac{11}{28}a-\frac{2}{7}$, $\frac{1}{28}a^{19}-\frac{3}{28}a^{16}-\frac{1}{28}a^{15}-\frac{1}{14}a^{14}+\frac{1}{14}a^{13}-\frac{5}{14}a^{12}-\frac{11}{28}a^{11}-\frac{3}{28}a^{10}-\frac{9}{28}a^{9}-\frac{1}{4}a^{8}-\frac{9}{28}a^{7}-\frac{1}{28}a^{6}+\frac{1}{14}a^{5}-\frac{9}{28}a^{4}+\frac{5}{14}a^{3}-\frac{1}{28}a^{2}+\frac{2}{7}a-\frac{13}{28}$, $\frac{1}{28}a^{20}+\frac{3}{28}a^{16}-\frac{3}{28}a^{15}-\frac{1}{14}a^{14}-\frac{3}{28}a^{13}+\frac{1}{4}a^{12}-\frac{2}{7}a^{11}+\frac{5}{28}a^{10}-\frac{5}{14}a^{9}-\frac{2}{7}a^{8}+\frac{5}{14}a^{7}+\frac{13}{28}a^{6}-\frac{2}{7}a^{5}+\frac{3}{28}a^{4}-\frac{2}{7}a^{3}-\frac{5}{28}a^{2}-\frac{3}{7}a+\frac{1}{7}$, $\frac{1}{28}a^{21}-\frac{1}{28}a^{15}+\frac{1}{28}a^{14}-\frac{3}{7}a^{12}+\frac{3}{28}a^{11}+\frac{11}{28}a^{10}+\frac{9}{28}a^{9}-\frac{3}{7}a^{8}-\frac{3}{7}a^{7}-\frac{3}{7}a^{6}-\frac{5}{28}a^{5}-\frac{1}{28}a^{4}+\frac{1}{14}a^{3}-\frac{13}{28}a^{2}-\frac{1}{7}a+\frac{5}{14}$, $\frac{1}{532}a^{22}-\frac{1}{76}a^{21}+\frac{9}{532}a^{20}+\frac{2}{133}a^{19}-\frac{1}{76}a^{18}+\frac{3}{532}a^{17}+\frac{12}{133}a^{16}+\frac{1}{76}a^{15}-\frac{17}{532}a^{14}+\frac{13}{266}a^{13}-\frac{47}{266}a^{12}+\frac{29}{76}a^{11}-\frac{17}{133}a^{10}-\frac{127}{266}a^{9}+\frac{85}{532}a^{8}-\frac{1}{133}a^{7}+\frac{31}{532}a^{6}-\frac{13}{38}a^{5}+\frac{101}{266}a^{4}-\frac{31}{133}a^{3}-\frac{15}{76}a^{2}-\frac{41}{266}a-\frac{25}{266}$, $\frac{1}{3724}a^{23}+\frac{3}{3724}a^{22}-\frac{61}{3724}a^{21}-\frac{27}{1862}a^{20}+\frac{27}{1862}a^{19}+\frac{1}{133}a^{18}-\frac{55}{3724}a^{17}+\frac{449}{3724}a^{16}+\frac{207}{1862}a^{15}-\frac{239}{3724}a^{14}+\frac{223}{3724}a^{13}-\frac{5}{38}a^{12}+\frac{639}{1862}a^{11}+\frac{1327}{3724}a^{10}-\frac{1847}{3724}a^{9}+\frac{979}{3724}a^{8}-\frac{617}{3724}a^{7}+\frac{59}{532}a^{6}+\frac{157}{532}a^{5}-\frac{591}{1862}a^{4}+\frac{555}{3724}a^{3}-\frac{1531}{3724}a^{2}-\frac{15}{3724}a+\frac{160}{931}$, $\frac{1}{1269884}a^{24}+\frac{3}{90706}a^{23}-\frac{3}{4123}a^{22}+\frac{130}{16709}a^{21}-\frac{11155}{634942}a^{20}+\frac{5067}{1269884}a^{19}+\frac{3483}{634942}a^{18}-\frac{45}{33418}a^{17}+\frac{11175}{317471}a^{16}+\frac{28496}{317471}a^{15}+\frac{950}{16709}a^{14}+\frac{30827}{317471}a^{13}+\frac{11475}{634942}a^{12}+\frac{70355}{317471}a^{11}-\frac{7951}{33418}a^{10}-\frac{74169}{1269884}a^{9}+\frac{138993}{634942}a^{8}+\frac{83180}{317471}a^{7}-\frac{59965}{181412}a^{6}+\frac{31125}{66836}a^{5}+\frac{543325}{1269884}a^{4}+\frac{133061}{634942}a^{3}-\frac{417}{5852}a^{2}+\frac{196125}{1269884}a+\frac{351265}{1269884}$, $\frac{1}{28\!\cdots\!96}a^{25}-\frac{24\!\cdots\!99}{28\!\cdots\!96}a^{24}-\frac{50\!\cdots\!67}{28\!\cdots\!96}a^{23}-\frac{13\!\cdots\!43}{28\!\cdots\!96}a^{22}-\frac{47\!\cdots\!77}{28\!\cdots\!96}a^{21}-\frac{55\!\cdots\!80}{72\!\cdots\!49}a^{20}-\frac{25\!\cdots\!09}{28\!\cdots\!96}a^{19}-\frac{70\!\cdots\!57}{28\!\cdots\!96}a^{18}-\frac{16\!\cdots\!03}{14\!\cdots\!98}a^{17}+\frac{31\!\cdots\!85}{28\!\cdots\!96}a^{16}+\frac{33\!\cdots\!35}{28\!\cdots\!96}a^{15}-\frac{62\!\cdots\!61}{28\!\cdots\!96}a^{14}-\frac{16\!\cdots\!85}{14\!\cdots\!98}a^{13}-\frac{25\!\cdots\!97}{72\!\cdots\!49}a^{12}+\frac{34\!\cdots\!87}{72\!\cdots\!49}a^{11}-\frac{57\!\cdots\!85}{18\!\cdots\!74}a^{10}+\frac{15\!\cdots\!73}{72\!\cdots\!49}a^{9}-\frac{46\!\cdots\!19}{72\!\cdots\!49}a^{8}+\frac{11\!\cdots\!33}{28\!\cdots\!96}a^{7}+\frac{16\!\cdots\!89}{46\!\cdots\!58}a^{6}-\frac{30\!\cdots\!11}{14\!\cdots\!98}a^{5}-\frac{80\!\cdots\!64}{72\!\cdots\!49}a^{4}-\frac{68\!\cdots\!43}{14\!\cdots\!98}a^{3}-\frac{15\!\cdots\!69}{41\!\cdots\!28}a^{2}+\frac{39\!\cdots\!51}{28\!\cdots\!96}a+\frac{35\!\cdots\!95}{72\!\cdots\!49}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $12$ | 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 solvable group of order 52 |
| The 16 conjugacy class representatives for $D_{26}$ |
| Character table for $D_{26}$ |
Intermediate fields
| \(\Q(\sqrt{-3247}) \), 13.1.48551226272641.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 26 sibling: | 26.2.23347285006439981249999555372409414171948497.1 |
| Minimal sibling: | 26.2.23347285006439981249999555372409414171948497.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/padicField/7.2.0.1}{2} }^{12}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{12}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.13.0.1}{13} }^{2}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{13}$ | $26$ | ${\href{/padicField/29.2.0.1}{2} }^{12}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{12}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{12}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.13.0.1}{13} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{13}$ | ${\href{/padicField/53.2.0.1}{2} }^{13}$ | ${\href{/padicField/59.13.0.1}{13} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| Deg $26$ | $2$ | $13$ | $13$ | |||
|
\(191\)
| 191.2.1.2 | $x^{2} + 191$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 191.2.1.2 | $x^{2} + 191$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 191.2.1.2 | $x^{2} + 191$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 191.2.1.2 | $x^{2} + 191$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 191.2.1.2 | $x^{2} + 191$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 191.2.1.2 | $x^{2} + 191$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 191.2.1.2 | $x^{2} + 191$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 191.2.1.2 | $x^{2} + 191$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 191.2.1.2 | $x^{2} + 191$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 191.2.1.2 | $x^{2} + 191$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 191.2.1.2 | $x^{2} + 191$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 191.2.1.2 | $x^{2} + 191$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 191.2.1.2 | $x^{2} + 191$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |