Normalized defining polynomial
\( x^{13} - 2x^{12} + x^{11} - 3x^{10} - 5x^{9} + 7x^{8} + 2x^{7} + 6x^{6} + 4x^{5} - 6x^{4} - 8x^{3} + 6x - 2 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[5, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(197683647463424\) \(\medspace = 2^{14}\cdot 12065652311\) | 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: | \(12.58\) | 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: | $2^{223/96}12065652311^{1/2}\approx 549595.2837715879$ | ||
Ramified primes: | \(2\), \(12065652311\) | 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: | $\Q(\sqrt{12065652311}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{12487}a^{12}-\frac{3467}{12487}a^{11}+\frac{662}{12487}a^{10}+\frac{3775}{12487}a^{9}+\frac{5996}{12487}a^{8}+\frac{2235}{12487}a^{7}-\frac{2333}{12487}a^{6}+\frac{4762}{12487}a^{5}-\frac{4999}{12487}a^{4}+\frac{2060}{12487}a^{3}+\frac{4656}{12487}a^{2}+\frac{164}{12487}a+\frac{6148}{12487}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $\frac{13309}{12487}a^{12}-\frac{27812}{12487}a^{11}+\frac{19710}{12487}a^{10}-\frac{56161}{12487}a^{9}-\frac{41114}{12487}a^{8}+\frac{64016}{12487}a^{7}+\frac{42733}{12487}a^{6}+\frac{105829}{12487}a^{5}+\frac{36506}{12487}a^{4}-\frac{42373}{12487}a^{3}-\frac{131147}{12487}a^{2}-\frac{15036}{12487}a+\frac{46369}{12487}$, $\frac{4013}{12487}a^{12}-\frac{15040}{12487}a^{11}+\frac{21849}{12487}a^{10}-\frac{35117}{12487}a^{9}+\frac{24473}{12487}a^{8}+\frac{28363}{12487}a^{7}-\frac{9566}{12487}a^{6}+\frac{42257}{12487}a^{5}-\frac{31839}{12487}a^{4}-\frac{24588}{12487}a^{3}-\frac{33485}{12487}a^{2}+\frac{21295}{12487}a-\frac{14875}{12487}$, $\frac{9042}{12487}a^{12}-\frac{18731}{12487}a^{11}+\frac{17018}{12487}a^{10}-\frac{43369}{12487}a^{9}-\frac{27696}{12487}a^{8}+\frac{29878}{12487}a^{7}+\frac{8044}{12487}a^{6}+\frac{77750}{12487}a^{5}+\frac{39443}{12487}a^{4}-\frac{4084}{12487}a^{3}-\frac{69047}{12487}a^{2}-\frac{3065}{12487}a+\frac{10579}{12487}$, $\frac{18988}{12487}a^{12}-\frac{24906}{12487}a^{11}-\frac{4353}{12487}a^{10}-\frac{45628}{12487}a^{9}-\frac{141775}{12487}a^{8}+\frac{69789}{12487}a^{7}+\frac{92281}{12487}a^{6}+\frac{164820}{12487}a^{5}+\frac{179980}{12487}a^{4}-\frac{31465}{12487}a^{3}-\frac{187137}{12487}a^{2}-\frac{132588}{12487}a+\frac{72183}{12487}$, $\frac{12906}{12487}a^{12}-\frac{29155}{12487}a^{11}+\frac{27638}{12487}a^{10}-\frac{66559}{12487}a^{9}-\frac{22537}{12487}a^{8}+\frac{49888}{12487}a^{7}+\frac{21433}{12487}a^{6}+\frac{109741}{12487}a^{5}+\frac{28209}{12487}a^{4}-\frac{23437}{12487}a^{3}-\frac{97004}{12487}a^{2}+\frac{6281}{12487}a+\frac{16177}{12487}$, $\frac{30030}{12487}a^{12}-\frac{47352}{12487}a^{11}+\frac{13043}{12487}a^{10}-\frac{93632}{12487}a^{9}-\frac{177478}{12487}a^{8}+\frac{111808}{12487}a^{7}+\frac{116950}{12487}a^{6}+\frac{238989}{12487}a^{5}+\frac{223510}{12487}a^{4}-\frac{61233}{12487}a^{3}-\frac{271973}{12487}a^{2}-\frac{119828}{12487}a+\frac{104041}{12487}$, $\frac{7269}{12487}a^{12}-\frac{15344}{12487}a^{11}+\frac{17070}{12487}a^{10}-\frac{43412}{12487}a^{9}-\frac{7193}{12487}a^{8}+\frac{628}{12487}a^{7}+\frac{11256}{12487}a^{6}+\frac{63449}{12487}a^{5}+\frac{11926}{12487}a^{4}+\frac{27201}{12487}a^{3}-\frac{57741}{12487}a^{2}+\frac{5851}{12487}a-\frac{1161}{12487}$, $\frac{17930}{12487}a^{12}-\frac{27998}{12487}a^{11}+\frac{7010}{12487}a^{10}-\frac{56225}{12487}a^{9}-\frac{104686}{12487}a^{8}+\frac{65202}{12487}a^{7}+\frac{75682}{12487}a^{6}+\frac{146398}{12487}a^{5}+\frac{124486}{12487}a^{4}-\frac{38207}{12487}a^{3}-\frac{180820}{12487}a^{2}-\frac{81334}{12487}a+\frac{60839}{12487}$ | 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: | \( 173.327894786 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{4}\cdot 173.327894786 \cdot 1}{2\cdot\sqrt{197683647463424}}\cr\approx \mathstrut & 0.307413223876 \end{aligned}\]
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 | R | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.13.0.1}{13} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
2.8.14.10 | $x^{8} + 2 x^{7} + 2 x^{2} + 4 x + 2$ | $8$ | $1$ | $14$ | $C_2 \wr S_4$ | $[4/3, 4/3, 2, 7/3, 7/3, 5/2]_{3}^{2}$ | |
\(12065652311\) | $\Q_{12065652311}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |