Normalized defining polynomial
\( x^{26} - 12 x^{24} + 54 x^{22} - 54 x^{20} + 891 x^{18} - 4617 x^{16} + 3645 x^{14} + 63423 x^{12} + \cdots - 1594323 \)
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: | \(252206691838729902820182233715746833170432\) \(\medspace = 2^{26}\cdot 3^{13}\cdot 191^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(39.12\) | 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: | $2\cdot 3^{1/2}191^{1/2}\approx 47.8748368143433$ | ||
Ramified primes: | \(2\), \(3\), \(191\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}$, $\frac{1}{9}a^{5}$, $\frac{1}{27}a^{6}$, $\frac{1}{27}a^{7}$, $\frac{1}{81}a^{8}$, $\frac{1}{81}a^{9}$, $\frac{1}{243}a^{10}$, $\frac{1}{243}a^{11}$, $\frac{1}{729}a^{12}$, $\frac{1}{729}a^{13}$, $\frac{1}{2187}a^{14}$, $\frac{1}{2187}a^{15}$, $\frac{1}{6561}a^{16}$, $\frac{1}{6561}a^{17}$, $\frac{1}{19683}a^{18}$, $\frac{1}{19683}a^{19}$, $\frac{1}{413343}a^{20}+\frac{1}{45927}a^{18}+\frac{1}{15309}a^{16}-\frac{1}{15309}a^{14}-\frac{1}{1701}a^{12}-\frac{1}{189}a^{8}-\frac{1}{189}a^{6}+\frac{2}{21}a^{2}-\frac{2}{7}$, $\frac{1}{413343}a^{21}+\frac{1}{45927}a^{19}+\frac{1}{15309}a^{17}-\frac{1}{15309}a^{15}-\frac{1}{1701}a^{13}-\frac{1}{189}a^{9}-\frac{1}{189}a^{7}+\frac{2}{21}a^{3}-\frac{2}{7}a$, $\frac{1}{1240029}a^{22}+\frac{1}{137781}a^{18}-\frac{1}{15309}a^{16}+\frac{2}{5103}a^{12}-\frac{1}{567}a^{10}+\frac{1}{567}a^{8}+\frac{1}{63}a^{6}+\frac{2}{63}a^{4}-\frac{1}{21}a^{2}-\frac{1}{7}$, $\frac{1}{1240029}a^{23}+\frac{1}{137781}a^{19}-\frac{1}{15309}a^{17}+\frac{2}{5103}a^{13}-\frac{1}{567}a^{11}+\frac{1}{567}a^{9}+\frac{1}{63}a^{7}+\frac{2}{63}a^{5}-\frac{1}{21}a^{3}-\frac{1}{7}a$, $\frac{1}{11\!\cdots\!03}a^{24}+\frac{144081863}{382539731844501}a^{22}-\frac{16226512}{42504414649389}a^{20}-\frac{144369395}{14168138216463}a^{18}-\frac{43879651}{14168138216463}a^{16}+\frac{711311152}{4722712738821}a^{14}-\frac{915385}{4727440179}a^{12}-\frac{813048479}{524745859869}a^{10}+\frac{1026038639}{174915286623}a^{8}-\frac{25093010}{2159447983}a^{6}-\frac{898360670}{19435031847}a^{4}+\frac{1049917522}{6478343949}a^{2}+\frac{940912370}{2159447983}$, $\frac{1}{11\!\cdots\!03}a^{25}+\frac{144081863}{382539731844501}a^{23}-\frac{16226512}{42504414649389}a^{21}-\frac{144369395}{14168138216463}a^{19}-\frac{43879651}{14168138216463}a^{17}+\frac{711311152}{4722712738821}a^{15}-\frac{915385}{4727440179}a^{13}-\frac{813048479}{524745859869}a^{11}+\frac{1026038639}{174915286623}a^{9}-\frac{25093010}{2159447983}a^{7}-\frac{898360670}{19435031847}a^{5}+\frac{1049917522}{6478343949}a^{3}+\frac{940912370}{2159447983}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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
\(\Q(\sqrt{3}) \), 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: | 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 | R | R | $26$ | ${\href{/padicField/7.2.0.1}{2} }^{13}$ | ${\href{/padicField/11.2.0.1}{2} }^{12}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/19.2.0.1}{2} }^{13}$ | ${\href{/padicField/23.13.0.1}{13} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{13}$ | ${\href{/padicField/31.2.0.1}{2} }^{13}$ | ${\href{/padicField/37.2.0.1}{2} }^{12}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{13}$ | $26$ | ${\href{/padicField/47.2.0.1}{2} }^{12}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{13}$ | ${\href{/padicField/59.13.0.1}{13} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $26$ | $2$ | $13$ | $26$ | |||
\(3\) | Deg $26$ | $2$ | $13$ | $13$ | |||
\(191\) | $\Q_{191}$ | $x + 172$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{191}$ | $x + 172$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 | $x^{2} + 191$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.2292.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 191 $ | \(\Q(\sqrt{-573}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.191.2t1.a.a | $1$ | $ 191 $ | \(\Q(\sqrt{-191}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 2.27504.26t3.a.d | $2$ | $ 2^{4} \cdot 3^{2} \cdot 191 $ | 26.2.252206691838729902820182233715746833170432.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |
* | 2.191.13t2.a.e | $2$ | $ 191 $ | 13.1.48551226272641.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.191.13t2.a.c | $2$ | $ 191 $ | 13.1.48551226272641.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.27504.26t3.a.e | $2$ | $ 2^{4} \cdot 3^{2} \cdot 191 $ | 26.2.252206691838729902820182233715746833170432.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |
* | 2.27504.26t3.a.f | $2$ | $ 2^{4} \cdot 3^{2} \cdot 191 $ | 26.2.252206691838729902820182233715746833170432.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |
* | 2.191.13t2.a.d | $2$ | $ 191 $ | 13.1.48551226272641.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.191.13t2.a.b | $2$ | $ 191 $ | 13.1.48551226272641.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.191.13t2.a.a | $2$ | $ 191 $ | 13.1.48551226272641.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.27504.26t3.a.a | $2$ | $ 2^{4} \cdot 3^{2} \cdot 191 $ | 26.2.252206691838729902820182233715746833170432.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |
* | 2.27504.26t3.a.c | $2$ | $ 2^{4} \cdot 3^{2} \cdot 191 $ | 26.2.252206691838729902820182233715746833170432.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |
* | 2.191.13t2.a.f | $2$ | $ 191 $ | 13.1.48551226272641.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.27504.26t3.a.b | $2$ | $ 2^{4} \cdot 3^{2} \cdot 191 $ | 26.2.252206691838729902820182233715746833170432.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |