Normalized defining polynomial
\( x^{13} - 2x^{12} + 2x^{10} - 3x^{9} + 4x^{8} - 4x^{7} + 4x^{6} - 5x^{5} + 3x^{4} - 2x^{3} + 3x^{2} - x - 1 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-14047799762079\) \(\medspace = -\,3^{3}\cdot 16103\cdot 32310059\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.26\) | 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: | $3^{1/2}16103^{1/2}32310059^{1/2}\approx 1249346.4852597937$ | ||
Ramified primes: | \(3\), \(16103\), \(32310059\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-1560866640231}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{279}a^{12}+\frac{2}{93}a^{11}+\frac{16}{93}a^{10}+\frac{107}{279}a^{9}+\frac{16}{279}a^{8}+\frac{44}{93}a^{7}-\frac{64}{279}a^{6}+\frac{50}{279}a^{5}+\frac{116}{279}a^{4}+\frac{94}{279}a^{3}-\frac{29}{93}a^{2}-\frac{15}{31}a+\frac{35}{279}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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)
| |
Fundamental units: | $\frac{25}{279}a^{12}-\frac{4}{31}a^{11}-\frac{1}{31}a^{10}-\frac{22}{279}a^{9}-\frac{65}{279}a^{8}+\frac{77}{93}a^{7}-\frac{205}{279}a^{6}+\frac{41}{279}a^{5}+\frac{110}{279}a^{4}+\frac{25}{279}a^{3}-\frac{4}{31}a^{2}+\frac{22}{93}a+\frac{224}{279}$, $a$, $\frac{55}{279}a^{12}-\frac{15}{31}a^{11}+\frac{4}{31}a^{10}+\frac{119}{279}a^{9}-\frac{143}{279}a^{8}+\frac{95}{93}a^{7}-\frac{451}{279}a^{6}+\frac{425}{279}a^{5}-\frac{316}{279}a^{4}+\frac{55}{279}a^{3}-\frac{15}{31}a^{2}+\frac{67}{93}a-\frac{121}{279}$, $\frac{13}{93}a^{12}-\frac{46}{93}a^{11}+\frac{4}{93}a^{10}+\frac{58}{93}a^{9}-\frac{3}{31}a^{8}+\frac{14}{31}a^{7}-\frac{88}{93}a^{6}+\frac{41}{31}a^{5}-\frac{166}{93}a^{4}+\frac{137}{93}a^{3}-\frac{77}{93}a^{2}+\frac{167}{93}a-\frac{24}{31}$, $\frac{38}{279}a^{12}-\frac{17}{93}a^{11}-\frac{43}{93}a^{10}+\frac{160}{279}a^{9}+\frac{50}{279}a^{8}-\frac{2}{93}a^{7}+\frac{79}{279}a^{6}-\frac{53}{279}a^{5}-\frac{56}{279}a^{4}-\frac{334}{279}a^{3}+\frac{14}{93}a^{2}-\frac{12}{31}a+\frac{214}{279}$, $\frac{1}{31}a^{12}-\frac{13}{93}a^{11}-\frac{11}{93}a^{10}+\frac{11}{93}a^{9}+\frac{17}{93}a^{8}+\frac{8}{31}a^{7}-\frac{2}{31}a^{6}+\frac{88}{93}a^{5}-\frac{8}{31}a^{4}-\frac{59}{93}a^{3}+\frac{49}{93}a^{2}+\frac{29}{93}a+\frac{43}{93}$, $\frac{46}{279}a^{12}-\frac{32}{93}a^{11}+\frac{23}{93}a^{10}+\frac{86}{279}a^{9}-\frac{194}{279}a^{8}+\frac{71}{93}a^{7}-\frac{433}{279}a^{6}+\frac{161}{279}a^{5}-\frac{244}{279}a^{4}+\frac{232}{279}a^{3}-\frac{94}{93}a^{2}+\frac{38}{93}a-\frac{250}{279}$ | 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: | \( 16.6395085824 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{5}\cdot 16.6395085824 \cdot 1}{2\cdot\sqrt{14047799762079}}\cr\approx \mathstrut & 0.173898435602 \end{aligned}\]
Galois group
A non-solvable group of order 6227020800 |
The 101 conjugacy class representatives for $S_{13}$ are not computed |
Character table for $S_{13}$ is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 | ${\href{/padicField/2.13.0.1}{13} }$ | R | ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.13.0.1}{13} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.13.0.1}{13} }$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.13.0.1}{13} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.13.0.1}{13} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(16103\) | $\Q_{16103}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
\(32310059\) | $\Q_{32310059}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{32310059}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |