Normalized defining polynomial
\( x^{26} - 3 \)
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: | \(5216009998678547223740865528327600202444877856768\) \(\medspace = 2^{26}\cdot 3^{25}\cdot 13^{26}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(74.77\) | 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: | $2\cdot 3^{25/26}13^{167/156}\approx 89.59650580978655$ | ||
Ramified primes: | \(2\), \(3\), \(13\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | $a^{13}-2$, $a^{14}-a^{2}-1$, $a^{25}-a^{24}+a^{23}-a^{22}+a^{21}-a^{20}+a^{19}-a^{18}+a^{17}-a^{16}+a^{15}-a^{14}+a^{13}-a^{12}+a^{11}-a^{10}+a^{9}-a^{8}+a^{7}-a^{6}+a^{5}-a^{4}+a^{3}-a^{2}+a-2$, $a^{22}+a^{20}+a^{16}+2a^{14}+a^{10}+2a^{8}+a^{6}+3a^{2}+1$, $2a^{24}-2a^{22}+2a^{20}-a^{18}+a^{16}-a^{12}+2a^{10}-3a^{8}+4a^{6}-5a^{4}+5a^{2}-7$, $a^{24}+a^{18}+a^{14}-a^{10}+a^{8}+2a^{4}+3a^{2}+1$, $a^{22}+a^{20}-a^{18}-2a^{16}+2a^{12}+a^{10}-a^{8}-a^{6}-1$, $a^{24}-a^{22}-a^{19}+a^{17}+a^{14}-a^{12}-a^{9}+a^{7}-a^{3}+a-1$, $a^{24}-a^{22}+2a^{20}-3a^{18}+3a^{16}-3a^{14}+4a^{12}-4a^{10}+3a^{8}-2a^{6}+2a^{4}-a^{2}-2$, $5a^{25}-4a^{24}-11a^{23}-5a^{22}-2a^{21}+9a^{20}+11a^{19}+6a^{18}-13a^{17}-13a^{16}-5a^{15}+10a^{14}+10a^{13}+13a^{12}-2a^{11}-15a^{10}-22a^{9}+4a^{8}+18a^{7}+20a^{6}-a^{5}-9a^{4}-25a^{3}-13a^{2}+9a+38$, $10a^{25}+7a^{24}-3a^{23}-17a^{22}-21a^{21}-29a^{20}-44a^{19}-46a^{18}-45a^{17}-53a^{16}-49a^{15}-37a^{14}-35a^{13}-27a^{12}-5a^{11}+9a^{10}+18a^{9}+43a^{8}+61a^{7}+61a^{6}+78a^{5}+92a^{4}+81a^{3}+80a^{2}+82a+56$, $2a^{25}-4a^{24}+a^{23}+3a^{22}+2a^{21}-9a^{20}+11a^{19}-a^{18}-7a^{17}+8a^{16}+a^{15}-2a^{14}-3a^{13}+11a^{12}-5a^{11}-6a^{10}+16a^{9}-10a^{8}+a^{7}+5a^{6}+2a^{5}-6a^{4}+a^{3}+17a^{2}-25a+16$, $8a^{25}+18a^{24}-7a^{23}-18a^{22}+2a^{21}+8a^{20}-9a^{19}+a^{18}+18a^{17}-7a^{16}-23a^{15}+18a^{14}+33a^{13}-14a^{12}-34a^{11}+6a^{10}+24a^{9}-11a^{8}-19a^{7}+13a^{6}+a^{5}-26a^{4}+22a^{3}+49a^{2}-22a-52$ (assuming GRH) | 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: | \( 228316605135874.6 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 228316605135874.6 \cdot 1}{2\cdot\sqrt{5216009998678547223740865528327600202444877856768}}\cr\approx \mathstrut & 0.756931722471413 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times F_{13}$ (as 26T10):
A solvable group of order 312 |
The 26 conjugacy class representatives for $C_2\times F_{13}$ |
Character table for $C_2\times F_{13}$ is not computed |
Intermediate fields
\(\Q(\sqrt{3}) \), 13.1.160960249522493526573.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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 | R | R | ${\href{/padicField/5.4.0.1}{4} }^{6}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.12.0.1}{12} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.12.0.1}{12} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.12.0.1}{12} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{6}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.12.0.1}{12} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $26$ | ${\href{/padicField/59.12.0.1}{12} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
Deg $24$ | $2$ | $12$ | $24$ | ||||
\(3\) | Deg $26$ | $26$ | $1$ | $25$ | |||
\(13\) | 13.13.13.1 | $x^{13} + 13 x + 13$ | $13$ | $1$ | $13$ | $F_{13}$ | $[13/12]_{12}$ |
13.13.13.1 | $x^{13} + 13 x + 13$ | $13$ | $1$ | $13$ | $F_{13}$ | $[13/12]_{12}$ |