Normalized defining polynomial
\( x^{26} - 5 \)
Invariants
Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(27338662801166956620931510927975177764892578125\) \(\medspace = 5^{25}\cdot 13^{26}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(61.10\) | 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))
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Galois root discriminant: | $5^{25/26}13^{167/156}\approx 73.21114221183463$ | ||
Ramified primes: | \(5\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{10}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{11}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{12}$
Monogenic: | Not computed | |
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)
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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)
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Fundamental units: | $\frac{1}{2}a^{13}-\frac{1}{2}$, $\frac{1}{2}a^{13}-a^{12}+a^{11}-a^{10}+a^{9}-a^{8}+a^{7}-a^{6}+a^{5}-a^{4}+a^{3}-a^{2}+a-\frac{1}{2}$, $\frac{1}{2}a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-a^{15}-a^{14}+\frac{1}{2}a^{10}+\frac{1}{2}a^{9}+a^{7}+a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{2}a^{3}+1$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{24}+\frac{1}{2}a^{22}-\frac{1}{2}a^{21}+a^{20}-a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}+\frac{1}{2}a^{15}-\frac{3}{2}a^{14}+a^{13}-\frac{3}{2}a^{12}+\frac{1}{2}a^{11}-\frac{1}{2}a^{9}+\frac{3}{2}a^{8}-2a^{7}+2a^{6}-\frac{3}{2}a^{5}+\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{5}{2}a^{2}+\frac{5}{2}a-3$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{22}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-a^{15}+a^{14}-\frac{1}{2}a^{10}+\frac{1}{2}a^{9}+a^{7}-a^{6}-\frac{1}{2}a^{5}+\frac{1}{2}a^{4}+\frac{1}{2}a^{3}-1$, $\frac{1}{2}a^{25}+\frac{1}{2}a^{24}+\frac{1}{2}a^{22}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{3}{2}a^{18}-a^{17}-a^{16}-\frac{3}{2}a^{15}-a^{14}-2a^{13}-\frac{3}{2}a^{12}-\frac{1}{2}a^{11}-a^{10}+\frac{1}{2}a^{9}+\frac{1}{2}a^{7}+\frac{5}{2}a^{6}+\frac{3}{2}a^{5}+3a^{4}+3a^{3}+\frac{5}{2}a^{2}+5a+3$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{23}+\frac{1}{2}a^{21}-a^{19}+a^{18}+\frac{1}{2}a^{17}-\frac{3}{2}a^{16}+\frac{1}{2}a^{15}+\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-a^{11}+\frac{5}{2}a^{10}-2a^{9}-\frac{1}{2}a^{8}+a^{7}+a^{5}-\frac{5}{2}a^{4}+\frac{3}{2}a^{3}+\frac{3}{2}a^{2}-3a+\frac{1}{2}$, $\frac{7}{2}a^{25}+3a^{24}+\frac{1}{2}a^{23}-\frac{5}{2}a^{22}-4a^{21}-4a^{20}-2a^{19}+\frac{3}{2}a^{18}+5a^{17}+6a^{16}+4a^{15}-\frac{1}{2}a^{14}-\frac{11}{2}a^{13}-\frac{15}{2}a^{12}-6a^{11}-\frac{3}{2}a^{10}+\frac{9}{2}a^{9}+10a^{8}+10a^{7}+4a^{6}-\frac{9}{2}a^{5}-11a^{4}-13a^{3}-8a^{2}+\frac{3}{2}a+\frac{23}{2}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{24}-a^{23}-a^{22}-a^{21}+\frac{1}{2}a^{20}+a^{19}+a^{18}+\frac{3}{2}a^{17}-a^{15}-\frac{3}{2}a^{14}-2a^{13}-\frac{1}{2}a^{12}+\frac{1}{2}a^{11}+2a^{10}+3a^{9}+a^{8}+\frac{1}{2}a^{7}-2a^{6}-4a^{5}-\frac{3}{2}a^{4}-a^{3}+2a^{2}+\frac{9}{2}a+3$, $a^{25}+\frac{3}{2}a^{24}-2a^{23}+\frac{1}{2}a^{22}+\frac{3}{2}a^{21}-\frac{3}{2}a^{20}-\frac{1}{2}a^{19}+\frac{3}{2}a^{18}-\frac{1}{2}a^{17}-a^{16}+a^{15}+a^{14}-a^{13}-a^{12}+\frac{5}{2}a^{11}-a^{10}-\frac{7}{2}a^{9}+\frac{9}{2}a^{8}+\frac{1}{2}a^{7}-\frac{11}{2}a^{6}+\frac{11}{2}a^{5}+\frac{5}{2}a^{4}-9a^{3}+5a^{2}+5a-11$, $2a^{25}-2a^{24}+\frac{3}{2}a^{23}-a^{22}-\frac{1}{2}a^{21}+2a^{20}-\frac{5}{2}a^{19}+3a^{18}-3a^{17}+\frac{5}{2}a^{16}-\frac{3}{2}a^{14}+3a^{13}-4a^{12}+5a^{11}-\frac{7}{2}a^{10}+a^{9}+\frac{1}{2}a^{8}-3a^{7}+\frac{11}{2}a^{6}-8a^{5}+5a^{4}-\frac{7}{2}a^{3}+\frac{5}{2}a-9$, $\frac{3}{2}a^{25}-a^{24}+\frac{3}{2}a^{22}-2a^{21}+2a^{19}-\frac{5}{2}a^{18}+\frac{1}{2}a^{17}+2a^{16}-3a^{15}+2a^{14}+\frac{3}{2}a^{13}-\frac{7}{2}a^{12}+3a^{11}+a^{10}-\frac{7}{2}a^{9}+3a^{8}-a^{7}-4a^{6}+\frac{11}{2}a^{5}-\frac{5}{2}a^{4}-5a^{3}+8a^{2}-3a-\frac{7}{2}$, $\frac{3}{2}a^{25}+\frac{9}{2}a^{24}+3a^{23}+5a^{22}+\frac{11}{2}a^{21}+4a^{20}+\frac{13}{2}a^{19}+4a^{18}+4a^{17}+4a^{16}-\frac{5}{2}a^{13}-\frac{15}{2}a^{12}-\frac{9}{2}a^{11}-12a^{10}-9a^{9}-\frac{21}{2}a^{8}-14a^{7}-\frac{17}{2}a^{6}-13a^{5}-9a^{4}-5a^{3}-5a^{2}+3a+\frac{17}{2}$ (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)]
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Regulator: | \( 9612138497230.408 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 9612138497230.408 \cdot 1}{2\cdot\sqrt{27338662801166956620931510927975177764892578125}}\cr\approx \mathstrut & 0.440169353206470 \end{aligned}\] (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
\(\Q(\sqrt{5}) \), 13.1.73944117820374267578125.1 |
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 | ${\href{/padicField/2.12.0.1}{12} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.12.0.1}{12} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.12.0.1}{12} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.12.0.1}{12} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.12.0.1}{12} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{6}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $26$ | ${\href{/padicField/59.12.0.1}{12} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | Deg $26$ | $26$ | $1$ | $25$ | |||
\(13\) | Deg $26$ | $13$ | $2$ | $26$ |