Normalized defining polynomial
\( x^{26} - x^{25} - 45 x^{24} + 44 x^{23} + 912 x^{22} - 836 x^{21} - 11055 x^{20} + 9149 x^{19} + \cdots - 1990843 \)
Invariants
Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(33169014017307639762747533311185137385697693\) \(\medspace = 13^{13}\cdot 263^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(47.19\) | 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: | $13^{1/2}263^{1/2}\approx 58.47221562417487$ | ||
Ramified primes: | \(13\), \(263\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{5}a^{19}+\frac{2}{5}a^{17}+\frac{1}{5}a^{16}-\frac{2}{5}a^{15}-\frac{2}{5}a^{14}+\frac{1}{5}a^{12}-\frac{1}{5}a^{11}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{35}a^{20}-\frac{2}{35}a^{19}-\frac{13}{35}a^{18}+\frac{12}{35}a^{17}+\frac{16}{35}a^{16}+\frac{2}{35}a^{15}-\frac{1}{35}a^{14}+\frac{1}{35}a^{13}+\frac{2}{35}a^{12}+\frac{3}{35}a^{11}-\frac{1}{35}a^{10}-\frac{17}{35}a^{9}-\frac{2}{35}a^{8}+\frac{2}{35}a^{6}-\frac{9}{35}a^{5}-\frac{17}{35}a^{4}+\frac{16}{35}a^{3}-\frac{3}{35}a-\frac{13}{35}$, $\frac{1}{35}a^{21}-\frac{3}{35}a^{19}-\frac{2}{5}a^{18}-\frac{2}{35}a^{17}+\frac{13}{35}a^{16}+\frac{2}{7}a^{15}+\frac{6}{35}a^{14}+\frac{4}{35}a^{13}-\frac{2}{5}a^{12}-\frac{9}{35}a^{11}-\frac{1}{7}a^{10}+\frac{13}{35}a^{9}-\frac{4}{35}a^{8}+\frac{9}{35}a^{7}+\frac{9}{35}a^{6}-\frac{2}{5}a^{5}+\frac{3}{35}a^{4}+\frac{11}{35}a^{3}+\frac{4}{35}a^{2}-\frac{1}{7}a-\frac{1}{7}$, $\frac{1}{175}a^{22}+\frac{1}{175}a^{21}+\frac{1}{175}a^{20}+\frac{2}{35}a^{19}-\frac{33}{175}a^{18}-\frac{46}{175}a^{17}-\frac{53}{175}a^{16}-\frac{81}{175}a^{15}-\frac{64}{175}a^{14}-\frac{6}{175}a^{13}+\frac{11}{35}a^{12}-\frac{37}{175}a^{11}+\frac{74}{175}a^{10}-\frac{59}{175}a^{9}-\frac{38}{175}a^{8}-\frac{52}{175}a^{7}-\frac{32}{175}a^{6}+\frac{58}{175}a^{5}+\frac{86}{175}a^{4}-\frac{61}{175}a^{3}-\frac{1}{175}a^{2}+\frac{83}{175}a+\frac{83}{175}$, $\frac{1}{3325}a^{23}+\frac{4}{3325}a^{22}+\frac{24}{3325}a^{21}+\frac{8}{3325}a^{20}+\frac{17}{3325}a^{19}-\frac{2}{665}a^{18}+\frac{1599}{3325}a^{17}+\frac{177}{665}a^{16}-\frac{607}{3325}a^{15}+\frac{662}{3325}a^{14}+\frac{116}{475}a^{13}-\frac{617}{3325}a^{12}+\frac{398}{3325}a^{11}+\frac{1188}{3325}a^{10}-\frac{13}{133}a^{9}-\frac{1461}{3325}a^{8}+\frac{27}{3325}a^{7}+\frac{377}{3325}a^{6}+\frac{96}{665}a^{5}+\frac{1497}{3325}a^{4}+\frac{26}{3325}a^{3}+\frac{214}{665}a^{2}-\frac{383}{3325}a+\frac{144}{3325}$, $\frac{1}{9679075}a^{24}-\frac{118}{1935815}a^{23}+\frac{15394}{9679075}a^{22}+\frac{12333}{9679075}a^{21}+\frac{91291}{9679075}a^{20}-\frac{27287}{509425}a^{19}-\frac{193234}{9679075}a^{18}+\frac{1459348}{9679075}a^{17}-\frac{504818}{1935815}a^{16}+\frac{1683734}{9679075}a^{15}+\frac{140209}{387163}a^{14}+\frac{2167574}{9679075}a^{13}-\frac{1730419}{9679075}a^{12}+\frac{4035539}{9679075}a^{11}+\frac{4035187}{9679075}a^{10}-\frac{685719}{1935815}a^{9}+\frac{4822863}{9679075}a^{8}+\frac{3172292}{9679075}a^{7}-\frac{410434}{1935815}a^{6}+\frac{23415}{55309}a^{5}+\frac{134699}{9679075}a^{4}+\frac{684834}{1935815}a^{3}+\frac{932411}{9679075}a^{2}-\frac{3859216}{9679075}a-\frac{4049468}{9679075}$, $\frac{1}{81\!\cdots\!75}a^{25}-\frac{30\!\cdots\!51}{61\!\cdots\!75}a^{24}+\frac{99\!\cdots\!46}{81\!\cdots\!75}a^{23}-\frac{23\!\cdots\!49}{11\!\cdots\!25}a^{22}+\frac{74\!\cdots\!33}{81\!\cdots\!75}a^{21}+\frac{48\!\cdots\!47}{43\!\cdots\!25}a^{20}+\frac{46\!\cdots\!49}{81\!\cdots\!75}a^{19}+\frac{29\!\cdots\!71}{81\!\cdots\!75}a^{18}-\frac{31\!\cdots\!89}{81\!\cdots\!75}a^{17}-\frac{40\!\cdots\!82}{16\!\cdots\!75}a^{16}-\frac{37\!\cdots\!34}{81\!\cdots\!75}a^{15}-\frac{38\!\cdots\!59}{81\!\cdots\!75}a^{14}+\frac{74\!\cdots\!88}{16\!\cdots\!75}a^{13}+\frac{49\!\cdots\!37}{81\!\cdots\!75}a^{12}+\frac{18\!\cdots\!28}{16\!\cdots\!75}a^{11}+\frac{16\!\cdots\!52}{81\!\cdots\!75}a^{10}+\frac{34\!\cdots\!16}{81\!\cdots\!75}a^{9}-\frac{12\!\cdots\!38}{81\!\cdots\!75}a^{8}+\frac{89\!\cdots\!78}{43\!\cdots\!25}a^{7}-\frac{30\!\cdots\!92}{81\!\cdots\!75}a^{6}-\frac{24\!\cdots\!17}{81\!\cdots\!75}a^{5}-\frac{70\!\cdots\!51}{16\!\cdots\!75}a^{4}+\frac{85\!\cdots\!26}{23\!\cdots\!25}a^{3}-\frac{32\!\cdots\!92}{81\!\cdots\!75}a^{2}-\frac{30\!\cdots\!27}{81\!\cdots\!75}a-\frac{91\!\cdots\!12}{11\!\cdots\!25}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $5$ |
Class group and class number
not computed
Unit group
Rank: | $13$ | 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
\(\Q(\sqrt{13}) \), 13.1.330928743953809.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 26 sibling: | deg 26 |
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 | $26$ | ${\href{/padicField/3.13.0.1}{13} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{13}$ | ${\href{/padicField/7.2.0.1}{2} }^{13}$ | $26$ | R | ${\href{/padicField/17.13.0.1}{13} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{13}$ | ${\href{/padicField/23.13.0.1}{13} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $26$ | $26$ | ${\href{/padicField/41.2.0.1}{2} }^{13}$ | ${\href{/padicField/43.13.0.1}{13} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{13}$ | ${\href{/padicField/53.2.0.1}{2} }^{12}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{13}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | Deg $26$ | $2$ | $13$ | $13$ | |||
\(263\) | $\Q_{263}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{263}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |