Normalized defining polynomial
\( x^{13} - x^{12} - 8x^{11} + 3x^{10} + 28x^{9} - 50x^{7} - 3x^{6} + 44x^{5} - 4x^{4} - 17x^{3} + 5x^{2} + 2x - 1 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[9, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(748906923760081\) \(\medspace = 364027\cdot 2057284003\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.94\) | 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: | $364027^{1/2}2057284003^{1/2}\approx 27366163.848082196$ | ||
Ramified primes: | \(364027\), \(2057284003\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{748906923760081}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{67}a^{12}-\frac{23}{67}a^{11}+\frac{29}{67}a^{10}-\frac{32}{67}a^{9}-\frac{5}{67}a^{8}-\frac{24}{67}a^{7}+\frac{9}{67}a^{6}-\frac{23}{67}a^{4}+\frac{33}{67}a^{3}-\frac{6}{67}a^{2}+\frac{3}{67}a+\frac{3}{67}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $10$ | 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: | $a$, $\frac{100}{67}a^{12}-\frac{22}{67}a^{11}-\frac{852}{67}a^{10}-\frac{319}{67}a^{9}+\frac{2783}{67}a^{8}+\frac{2022}{67}a^{7}-\frac{4125}{67}a^{6}-50a^{5}+\frac{2859}{67}a^{4}+\frac{1558}{67}a^{3}-\frac{1136}{67}a^{2}-\frac{102}{67}a+\frac{166}{67}$, $a^{12}-a^{11}-7a^{10}+2a^{9}+21a^{8}+2a^{7}-29a^{6}-a^{5}+15a^{4}-5a^{3}-2a^{2}$, $\frac{58}{67}a^{12}+\frac{73}{67}a^{11}-\frac{529}{67}a^{10}-\frac{851}{67}a^{9}+\frac{1452}{67}a^{8}+\frac{3164}{67}a^{7}-\frac{1220}{67}a^{6}-68a^{5}+\frac{140}{67}a^{4}+\frac{2316}{67}a^{3}-\frac{482}{67}a^{2}-\frac{362}{67}a+\frac{241}{67}$, $\frac{124}{67}a^{12}-\frac{38}{67}a^{11}-\frac{1027}{67}a^{10}-\frac{350}{67}a^{9}+\frac{3333}{67}a^{8}+\frac{2384}{67}a^{7}-\frac{4914}{67}a^{6}-61a^{5}+\frac{3312}{67}a^{4}+\frac{2283}{67}a^{3}-\frac{1213}{67}a^{2}-\frac{432}{67}a+\frac{238}{67}$, $\frac{74}{67}a^{12}-\frac{27}{67}a^{11}-\frac{601}{67}a^{10}-\frac{157}{67}a^{9}+\frac{1908}{67}a^{8}+\frac{1172}{67}a^{7}-\frac{2818}{67}a^{6}-28a^{5}+\frac{1983}{67}a^{4}+\frac{901}{67}a^{3}-\frac{779}{67}a^{2}-\frac{180}{67}a+\frac{222}{67}$, $\frac{20}{67}a^{12}-\frac{58}{67}a^{11}-\frac{157}{67}a^{10}+\frac{365}{67}a^{9}+\frac{704}{67}a^{8}-\frac{882}{67}a^{7}-\frac{1696}{67}a^{6}+16a^{5}+\frac{1751}{67}a^{4}-\frac{747}{67}a^{3}-\frac{589}{67}a^{2}+\frac{194}{67}a+\frac{60}{67}$, $\frac{83}{67}a^{12}-\frac{33}{67}a^{11}-\frac{675}{67}a^{10}-\frac{177}{67}a^{9}+\frac{2198}{67}a^{8}+\frac{1358}{67}a^{7}-\frac{3340}{67}a^{6}-34a^{5}+\frac{2513}{67}a^{4}+\frac{1198}{67}a^{3}-\frac{1235}{67}a^{2}-\frac{220}{67}a+\frac{249}{67}$, $\frac{88}{67}a^{12}-\frac{81}{67}a^{11}-\frac{664}{67}a^{10}+\frac{132}{67}a^{9}+\frac{2173}{67}a^{8}+\frac{501}{67}a^{7}-\frac{3362}{67}a^{6}-17a^{5}+\frac{2063}{67}a^{4}+\frac{626}{67}a^{3}-\frac{260}{67}a^{2}-\frac{138}{67}a-\frac{4}{67}$, $\frac{78}{67}a^{12}+\frac{15}{67}a^{11}-\frac{686}{67}a^{10}-\frac{486}{67}a^{9}+\frac{2156}{67}a^{8}+\frac{2282}{67}a^{7}-\frac{2916}{67}a^{6}-52a^{5}+\frac{1891}{67}a^{4}+\frac{1569}{67}a^{3}-\frac{1004}{67}a^{2}-\frac{101}{67}a+\frac{234}{67}$ | 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: | \( 397.580907117 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{2}\cdot 397.580907117 \cdot 1}{2\cdot\sqrt{748906923760081}}\cr\approx \mathstrut & 0.146828816910 \end{aligned}\]
Galois group
A non-solvable group of order 6227020800 |
The 101 conjugacy class representatives for $S_{13}$ |
Character table for $S_{13}$ |
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 | ${\href{/padicField/2.13.0.1}{13} }$ | ${\href{/padicField/3.13.0.1}{13} }$ | ${\href{/padicField/5.13.0.1}{13} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.13.0.1}{13} }$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.3.0.1}{3} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(364027\) | $\Q_{364027}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
\(2057284003\) | $\Q_{2057284003}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |