Normalized defining polynomial
\( x^{26} - 3x^{13} + 3 \)
Invariants
Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-77724605779030132647467635994070771372987\) \(\medspace = -\,3^{25}\cdot 13^{26}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(37.39\) | 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: | $3^{25/26}13^{167/156}\approx 44.798252904893275$ | ||
Ramified primes: | \(3\), \(13\) | 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: | $12$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -a^{13} + 2 \) (order $6$) | 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: | $a^{14}-2a+1$, $a^{14}-a^{13}-a+1$, $a^{23}+a^{20}+a^{17}+a^{14}+a^{13}+a^{11}-a^{10}+a^{8}-a^{7}+a^{5}-a^{4}+a^{2}-a-1$, $a^{25}+a^{24}+a^{23}+a^{22}-a^{19}-a^{18}-a^{17}-a^{16}+a^{13}-a^{12}-a^{11}-2a^{10}-2a^{9}-a^{8}-a^{7}+a^{6}+a^{5}+2a^{4}+2a^{3}+a^{2}+a-1$, $a^{24}-a^{23}-a^{22}+a^{21}-a^{19}+a^{17}-a^{16}+2a^{14}-a^{13}-a^{12}-2a^{11}+2a^{10}+a^{9}-2a^{8}+a^{7}+a^{6}-a^{5}+2a^{3}-a^{2}-3a+2$, $a^{23}-a^{22}+a^{21}-a^{19}+2a^{18}-a^{17}+a^{16}-a^{14}+a^{13}+a^{12}-a^{11}-a^{10}+a^{9}-2a^{8}+a^{7}+a^{6}-3a^{5}+2a^{4}-3a^{3}+a^{2}+a-2$, $a^{25}-a^{24}+a^{23}+a^{18}-a^{17}+2a^{16}-a^{15}+a^{14}-a^{13}-a^{12}+2a^{11}-2a^{10}+a^{9}-a^{8}+a^{7}-a^{6}+a^{4}-3a^{3}+2a^{2}-2a+2$, $a^{21}-2a^{20}+2a^{19}-a^{18}+a^{17}-2a^{16}+2a^{15}-a^{14}+a^{12}-a^{11}+a^{10}-a^{9}+2a^{7}-3a^{6}+2a^{5}-2a^{4}+3a^{3}-4a^{2}+4a-2$, $a^{24}+a^{22}+a^{21}-a^{18}-a^{17}-2a^{16}-a^{14}-2a^{11}-a^{9}-a^{8}+a^{7}+a^{6}+3a^{5}+a^{4}+4a^{3}-a^{2}+a-1$, $a^{25}-a^{24}+2a^{23}-a^{22}+2a^{20}-2a^{19}+2a^{18}+3a^{15}-2a^{14}+2a^{13}-2a^{12}+2a^{11}-3a^{10}+a^{9}+a^{8}-4a^{7}+3a^{6}-3a^{5}-a^{4}+a^{3}-6a^{2}+3a-2$, $a^{24}+a^{22}-a^{21}+a^{20}-a^{19}+a^{18}-2a^{17}+a^{16}-a^{15}+a^{14}-a^{13}+a^{12}-2a^{11}+a^{10}-2a^{9}+2a^{8}-2a^{7}+2a^{6}-a^{5}+3a^{4}-a^{3}+a^{2}+1$, $a^{21}-a^{19}-a^{18}+a^{17}+2a^{16}-a^{15}-a^{14}+a^{13}-a^{11}+a^{9}-a^{8}-a^{7}+a^{6}+3a^{5}-2a^{4}-4a^{3}+2a^{2}+2a-1$ (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: | \( 43085229111.75138 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 43085229111.75138 \cdot 1}{6\cdot\sqrt{77724605779030132647467635994070771372987}}\cr\approx \mathstrut & 0.612683494871764 \end{aligned}\] (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 $C_2\times F_{13}$ |
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 | ${\href{/padicField/2.12.0.1}{12} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.4.0.1}{4} }^{6}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.12.0.1}{12} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.6.0.1}{6} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.12.0.1}{12} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{6}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $26$ | ${\href{/padicField/59.12.0.1}{12} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 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 | $x^{13} + 13 x + 13$ | $13$ | $1$ | $13$ | $F_{13}$ | $[13/12]_{12}$ |