Normalized defining polynomial
\( x^{13} - 3 x^{12} + 3 x^{11} - 3 x^{10} + 5 x^{9} + 2 x^{8} - 17 x^{7} + 10 x^{6} + 16 x^{5} - 16 x^{4} + \cdots - 1 \)
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)
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Discriminant: | \(-69188908404875\) \(\medspace = -\,5^{3}\cdot 7^{5}\cdot 31\cdot 1062367\) | 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: | \(11.60\) | 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: | $5^{3/4}7^{5/6}31^{1/2}1062367^{1/2}\approx 97116.7192341759$ | ||
Ramified primes: | \(5\), \(7\), \(31\), \(1062367\) | 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: | $\Q(\sqrt{-1152668195}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{1163}a^{12}-\frac{16}{1163}a^{11}+\frac{211}{1163}a^{10}-\frac{420}{1163}a^{9}-\frac{350}{1163}a^{8}-\frac{100}{1163}a^{7}+\frac{120}{1163}a^{6}-\frac{387}{1163}a^{5}+\frac{395}{1163}a^{4}-\frac{499}{1163}a^{3}-\frac{496}{1163}a^{2}-\frac{523}{1163}a-\frac{179}{1163}$
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)
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Fundamental units: | $\frac{26}{1163}a^{12}+\frac{747}{1163}a^{11}-\frac{2655}{1163}a^{10}+\frac{3036}{1163}a^{9}-\frac{3285}{1163}a^{8}+\frac{5541}{1163}a^{7}-\frac{369}{1163}a^{6}-\frac{13551}{1163}a^{5}+\frac{11433}{1163}a^{4}+\frac{10286}{1163}a^{3}-\frac{12896}{1163}a^{2}-\frac{1968}{1163}a+\frac{3487}{1163}$, $a$, $\frac{1611}{1163}a^{12}-\frac{4842}{1163}a^{11}+\frac{4977}{1163}a^{10}-\frac{5569}{1163}a^{9}+\frac{9509}{1163}a^{8}+\frac{1720}{1163}a^{7}-\frac{25324}{1163}a^{6}+\frac{13867}{1163}a^{5}+\frac{23444}{1163}a^{4}-\frac{20027}{1163}a^{3}-\frac{8216}{1163}a^{2}+\frac{5274}{1163}a+\frac{1218}{1163}$, $\frac{790}{1163}a^{12}-\frac{3336}{1163}a^{11}+\frac{5033}{1163}a^{10}-\frac{4997}{1163}a^{9}+\frac{7272}{1163}a^{8}-\frac{3405}{1163}a^{7}-\frac{15685}{1163}a^{6}+\frac{22236}{1163}a^{5}+\frac{5018}{1163}a^{4}-\frac{23213}{1163}a^{3}+\frac{5906}{1163}a^{2}+\frac{6673}{1163}a-\frac{1850}{1163}$, $\frac{1060}{1163}a^{12}-\frac{3004}{1163}a^{11}+\frac{2690}{1163}a^{10}-\frac{3260}{1163}a^{9}+\frac{5812}{1163}a^{8}+\frac{2159}{1163}a^{7}-\frac{15849}{1163}a^{6}+\frac{6134}{1163}a^{5}+\frac{16302}{1163}a^{4}-\frac{9079}{1163}a^{3}-\frac{8225}{1163}a^{2}+\frac{2697}{1163}a+\frac{2155}{1163}$, $\frac{2228}{1163}a^{12}-\frac{6573}{1163}a^{11}+\frac{7234}{1163}a^{10}-\frac{8849}{1163}a^{9}+\frac{13366}{1163}a^{8}+\frac{1659}{1163}a^{7}-\frac{32694}{1163}a^{6}+\frac{21644}{1163}a^{5}+\frac{24092}{1163}a^{4}-\frac{26693}{1163}a^{3}-\frac{3727}{1163}a^{2}+\frac{7060}{1163}a-\frac{1066}{1163}$, $\frac{1034}{1163}a^{12}-\frac{3751}{1163}a^{11}+\frac{5345}{1163}a^{10}-\frac{6296}{1163}a^{9}+\frac{9097}{1163}a^{8}-\frac{3382}{1163}a^{7}-\frac{15480}{1163}a^{6}+\frac{19685}{1163}a^{5}+\frac{4869}{1163}a^{4}-\frac{19365}{1163}a^{3}+\frac{4671}{1163}a^{2}+\frac{4665}{1163}a-\frac{1332}{1163}$ | 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: | \( 49.3240960032 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{5}\cdot 49.3240960032 \cdot 1}{2\cdot\sqrt{69188908404875}}\cr\approx \mathstrut & 0.232273801635 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.13.0.1}{13} }$ | R | R | ${\href{/padicField/11.13.0.1}{13} }$ | ${\href{/padicField/13.13.0.1}{13} }$ | ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.13.0.1}{13} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.13.0.1}{13} }$ | R | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.13.0.1}{13} }$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.9.0.1 | $x^{9} + 2 x^{3} + x + 3$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
\(7\) | 7.6.5.1 | $x^{6} + 21$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
7.7.0.1 | $x^{7} + 6 x + 4$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(31\) | $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.5.0.1 | $x^{5} + 7 x + 28$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
31.5.0.1 | $x^{5} + 7 x + 28$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
\(1062367\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |