Normalized defining polynomial
\( x^{13} - x^{12} + x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + 4x^{5} - 5x^{4} + 4x^{3} - 2x + 1 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1414971204906609\) \(\medspace = 3^{6}\cdot 7^{2}\cdot 29^{2}\cdot 6863^{2}\) | 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: | \(14.64\) | 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: | $3^{6/7}7^{1/2}29^{2/3}6863^{1/2}\approx 5305.167415671284$ | ||
Ramified primes: | \(3\), \(7\), \(29\), \(6863\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{29}a^{12}+\frac{14}{29}a^{11}+\frac{8}{29}a^{10}+\frac{5}{29}a^{9}-\frac{13}{29}a^{8}+\frac{9}{29}a^{7}-\frac{11}{29}a^{6}+\frac{9}{29}a^{5}-\frac{6}{29}a^{4}-\frac{8}{29}a^{3}-\frac{2}{29}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $a$, $\frac{13}{29}a^{12}-\frac{21}{29}a^{11}+\frac{17}{29}a^{10}+\frac{7}{29}a^{9}-\frac{24}{29}a^{8}+\frac{1}{29}a^{7}-\frac{27}{29}a^{6}+\frac{1}{29}a^{5}+\frac{38}{29}a^{4}-\frac{104}{29}a^{3}+3a^{2}-\frac{26}{29}$, $\frac{6}{29}a^{12}-\frac{3}{29}a^{11}+\frac{19}{29}a^{10}+\frac{1}{29}a^{9}+\frac{9}{29}a^{8}+\frac{25}{29}a^{7}-\frac{8}{29}a^{6}-\frac{4}{29}a^{5}+\frac{22}{29}a^{4}-\frac{19}{29}a^{3}+2a^{2}-a+\frac{17}{29}$, $\frac{22}{29}a^{12}-\frac{11}{29}a^{11}+\frac{2}{29}a^{10}+\frac{23}{29}a^{9}-\frac{25}{29}a^{8}-\frac{5}{29}a^{7}-\frac{39}{29}a^{6}-\frac{34}{29}a^{5}+\frac{100}{29}a^{4}-\frac{60}{29}a^{3}+a^{2}+2a-\frac{44}{29}$, $\frac{23}{29}a^{12}+\frac{3}{29}a^{11}+\frac{10}{29}a^{10}+\frac{28}{29}a^{9}+\frac{20}{29}a^{8}+\frac{4}{29}a^{7}-\frac{21}{29}a^{6}-\frac{25}{29}a^{5}+\frac{65}{29}a^{4}-\frac{10}{29}a^{3}+\frac{12}{29}$, $\frac{5}{29}a^{12}+\frac{12}{29}a^{11}-\frac{18}{29}a^{10}+\frac{25}{29}a^{9}-\frac{7}{29}a^{8}-\frac{13}{29}a^{7}+\frac{3}{29}a^{6}-\frac{42}{29}a^{5}+\frac{28}{29}a^{4}+\frac{18}{29}a^{3}-2a^{2}+3a-\frac{39}{29}$ | 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: | \( 307.196082794 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{6}\cdot 307.196082794 \cdot 1}{2\cdot\sqrt{1414971204906609}}\cr\approx \mathstrut & 0.502482663758 \end{aligned}\]
Galois group
A non-solvable group of order 3113510400 |
The 55 conjugacy class representatives for $A_{13}$ |
Character table for $A_{13}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.13.0.1}{13} }$ | R | ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.13.0.1}{13} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | R | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.13.0.1}{13} }$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.13.0.1}{13} }$ | ${\href{/padicField/53.13.0.1}{13} }$ | ${\href{/padicField/59.13.0.1}{13} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
3.7.6.1 | $x^{7} + 3$ | $7$ | $1$ | $6$ | $F_7$ | $[\ ]_{7}^{6}$ | |
\(7\) | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
7.9.0.1 | $x^{9} + 6 x^{4} + x^{3} + 6 x + 4$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
\(29\) | $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
29.3.2.1 | $x^{3} + 29$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
29.6.0.1 | $x^{6} + x^{4} + 25 x^{3} + 17 x^{2} + 13 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(6863\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |