Normalized defining polynomial
\( x^{26} - 2 x^{25} - 45 x^{24} + 178 x^{23} + 768 x^{22} - 5448 x^{21} + 801 x^{20} + 65886 x^{19} - 134835 x^{18} - 349218 x^{17} + 1833359 x^{16} - 1872300 x^{15} + \cdots + 8027125537 \)
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: | \(-1919641021107487828653877081110544587155912070949008048128\) \(\medspace = -\,2^{39}\cdot 79^{24}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(159.66\) | 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: | $2^{3/2}79^{12/13}\approx 159.66170770416778$ | ||
Ramified primes: | \(2\), \(79\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
$\card{ \Gal(K/\Q) }$: | $26$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(632=2^{3}\cdot 79\) | ||
Dirichlet character group: | $\lbrace$$\chi_{632}(1,·)$, $\chi_{632}(259,·)$, $\chi_{632}(97,·)$, $\chi_{632}(179,·)$, $\chi_{632}(65,·)$, $\chi_{632}(275,·)$, $\chi_{632}(457,·)$, $\chi_{632}(459,·)$, $\chi_{632}(337,·)$, $\chi_{632}(403,·)$, $\chi_{632}(225,·)$, $\chi_{632}(89,·)$, $\chi_{632}(539,·)$, $\chi_{632}(289,·)$, $\chi_{632}(67,·)$, $\chi_{632}(283,·)$, $\chi_{632}(131,·)$, $\chi_{632}(561,·)$, $\chi_{632}(617,·)$, $\chi_{632}(299,·)$, $\chi_{632}(433,·)$, $\chi_{632}(563,·)$, $\chi_{632}(417,·)$, $\chi_{632}(441,·)$, $\chi_{632}(571,·)$, $\chi_{632}(475,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{4096}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{23}a^{20}+\frac{1}{23}a^{19}+\frac{6}{23}a^{18}+\frac{1}{23}a^{17}+\frac{8}{23}a^{16}-\frac{5}{23}a^{15}+\frac{2}{23}a^{14}-\frac{10}{23}a^{13}+\frac{10}{23}a^{12}+\frac{10}{23}a^{11}-\frac{5}{23}a^{10}+\frac{5}{23}a^{8}-\frac{11}{23}a^{7}+\frac{2}{23}a^{6}+\frac{7}{23}a^{5}-\frac{11}{23}a^{4}+\frac{6}{23}a^{3}-\frac{9}{23}a^{2}+\frac{3}{23}a-\frac{6}{23}$, $\frac{1}{23}a^{21}+\frac{5}{23}a^{19}-\frac{5}{23}a^{18}+\frac{7}{23}a^{17}+\frac{10}{23}a^{16}+\frac{7}{23}a^{15}+\frac{11}{23}a^{14}-\frac{3}{23}a^{13}+\frac{8}{23}a^{11}+\frac{5}{23}a^{10}+\frac{5}{23}a^{9}+\frac{7}{23}a^{8}-\frac{10}{23}a^{7}+\frac{5}{23}a^{6}+\frac{5}{23}a^{5}-\frac{6}{23}a^{4}+\frac{8}{23}a^{3}-\frac{11}{23}a^{2}-\frac{9}{23}a+\frac{6}{23}$, $\frac{1}{23}a^{22}-\frac{10}{23}a^{19}+\frac{5}{23}a^{17}-\frac{10}{23}a^{16}-\frac{10}{23}a^{15}+\frac{10}{23}a^{14}+\frac{4}{23}a^{13}+\frac{4}{23}a^{12}+\frac{1}{23}a^{11}+\frac{7}{23}a^{10}+\frac{7}{23}a^{9}+\frac{11}{23}a^{8}-\frac{9}{23}a^{7}-\frac{5}{23}a^{6}+\frac{5}{23}a^{5}-\frac{6}{23}a^{4}+\frac{5}{23}a^{3}-\frac{10}{23}a^{2}-\frac{9}{23}a+\frac{7}{23}$, $\frac{1}{23}a^{23}+\frac{10}{23}a^{19}-\frac{4}{23}a^{18}+\frac{1}{23}a^{16}+\frac{6}{23}a^{15}+\frac{1}{23}a^{14}-\frac{4}{23}a^{13}+\frac{9}{23}a^{12}-\frac{8}{23}a^{11}+\frac{3}{23}a^{10}+\frac{11}{23}a^{9}-\frac{5}{23}a^{8}+\frac{2}{23}a^{6}-\frac{5}{23}a^{5}+\frac{10}{23}a^{4}+\frac{4}{23}a^{3}-\frac{7}{23}a^{2}-\frac{9}{23}a+\frac{9}{23}$, $\frac{1}{1253201}a^{24}-\frac{22033}{1253201}a^{23}+\frac{26467}{1253201}a^{22}+\frac{16710}{1253201}a^{21}-\frac{22941}{1253201}a^{20}+\frac{272495}{1253201}a^{19}-\frac{637}{2369}a^{18}+\frac{469626}{1253201}a^{17}+\frac{303}{1253201}a^{16}-\frac{80824}{1253201}a^{15}+\frac{334265}{1253201}a^{14}-\frac{260376}{1253201}a^{13}+\frac{515943}{1253201}a^{12}+\frac{14986}{54487}a^{11}-\frac{383738}{1253201}a^{10}-\frac{461126}{1253201}a^{9}+\frac{130323}{1253201}a^{8}-\frac{46281}{1253201}a^{7}+\frac{430331}{1253201}a^{6}-\frac{104341}{1253201}a^{5}+\frac{89829}{1253201}a^{4}+\frac{518892}{1253201}a^{3}-\frac{10219}{54487}a^{2}-\frac{339893}{1253201}a-\frac{150560}{1253201}$, $\frac{1}{23\!\cdots\!53}a^{25}-\frac{75\!\cdots\!21}{23\!\cdots\!53}a^{24}-\frac{30\!\cdots\!43}{23\!\cdots\!53}a^{23}+\frac{65\!\cdots\!09}{23\!\cdots\!51}a^{22}-\frac{67\!\cdots\!62}{23\!\cdots\!53}a^{21}-\frac{42\!\cdots\!53}{23\!\cdots\!53}a^{20}-\frac{80\!\cdots\!55}{23\!\cdots\!53}a^{19}+\frac{61\!\cdots\!97}{23\!\cdots\!53}a^{18}+\frac{60\!\cdots\!73}{23\!\cdots\!53}a^{17}+\frac{17\!\cdots\!83}{23\!\cdots\!53}a^{16}+\frac{75\!\cdots\!91}{23\!\cdots\!53}a^{15}+\frac{13\!\cdots\!15}{10\!\cdots\!11}a^{14}-\frac{82\!\cdots\!94}{23\!\cdots\!53}a^{13}-\frac{83\!\cdots\!58}{23\!\cdots\!53}a^{12}+\frac{56\!\cdots\!00}{23\!\cdots\!53}a^{11}-\frac{39\!\cdots\!18}{23\!\cdots\!53}a^{10}+\frac{43\!\cdots\!83}{10\!\cdots\!11}a^{9}+\frac{10\!\cdots\!44}{23\!\cdots\!53}a^{8}+\frac{10\!\cdots\!13}{23\!\cdots\!53}a^{7}-\frac{88\!\cdots\!44}{23\!\cdots\!53}a^{6}-\frac{53\!\cdots\!73}{23\!\cdots\!53}a^{5}-\frac{16\!\cdots\!71}{23\!\cdots\!53}a^{4}+\frac{28\!\cdots\!58}{23\!\cdots\!53}a^{3}+\frac{98\!\cdots\!41}{23\!\cdots\!53}a^{2}-\frac{24\!\cdots\!66}{23\!\cdots\!53}a+\frac{99\!\cdots\!24}{23\!\cdots\!53}$
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 cyclic group of order 26 |
The 26 conjugacy class representatives for $C_{26}$ |
Character table for $C_{26}$ is not computed |
Intermediate fields
\(\Q(\sqrt{-2}) \), 13.13.59091511031674153381441.1 |
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 | R | ${\href{/padicField/3.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/padicField/11.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/17.13.0.1}{13} }^{2}$ | ${\href{/padicField/19.13.0.1}{13} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{13}$ | $26$ | $26$ | $26$ | ${\href{/padicField/41.13.0.1}{13} }^{2}$ | ${\href{/padicField/43.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/padicField/59.13.0.1}{13} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $26$ | $2$ | $13$ | $39$ | |||
\(79\) | Deg $26$ | $13$ | $2$ | $24$ |