Normalized defining polynomial
\( x^{13} - 6 x^{11} - 3 x^{10} + 9 x^{9} + 19 x^{8} + 7 x^{7} - 42 x^{6} - 25 x^{5} + 39 x^{4} + 15 x^{3} + \cdots + 1 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[11, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-4712558875285687\) | 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: | \(16.06\) | 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: | $4712558875285687^{1/2}\approx 68648079.9096791$ | ||
Ramified primes: | \(4712558875285687\) | 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: | $\Q(\sqrt{-47125\!\cdots\!85687}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $11$ | 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: | $a^{12}+3a^{11}-7a^{10}-18a^{9}+3a^{8}+35a^{7}+58a^{6}-22a^{5}-115a^{4}-3a^{3}+60a^{2}+3a-6$, $a$, $2a^{12}+5a^{11}-13a^{10}-30a^{9}+5a^{8}+59a^{7}+98a^{6}-41a^{5}-182a^{4}+2a^{3}+85a^{2}+4a-6$, $5a^{12}+5a^{11}-28a^{10}-41a^{9}+18a^{8}+114a^{7}+134a^{6}-120a^{5}-250a^{4}+33a^{3}+114a^{2}-6$, $7a^{12}+8a^{11}-39a^{10}-63a^{9}+21a^{8}+166a^{7}+207a^{6}-155a^{5}-383a^{4}+17a^{3}+182a^{2}+12a-14$, $6a^{12}+8a^{11}-34a^{10}-59a^{9}+16a^{8}+146a^{7}+196a^{6}-126a^{5}-354a^{4}+4a^{3}+160a^{2}+16a-11$, $11a^{12}+7a^{11}-59a^{10}-72a^{9}+41a^{8}+233a^{7}+238a^{6}-271a^{5}-439a^{4}+75a^{3}+199a^{2}+7a-14$, $9a^{12}+6a^{11}-48a^{10}-60a^{9}+32a^{8}+190a^{7}+198a^{6}-216a^{5}-362a^{4}+57a^{3}+161a^{2}+6a-9$, $3a^{11}-a^{10}-15a^{9}-6a^{8}+16a^{7}+51a^{6}+20a^{5}-90a^{4}-42a^{3}+45a^{2}+16a-4$, $13a^{12}+8a^{11}-69a^{10}-84a^{9}+46a^{8}+273a^{7}+280a^{6}-312a^{5}-508a^{4}+73a^{3}+224a^{2}+17a-13$, $a^{12}+a^{11}-6a^{10}-8a^{9}+5a^{8}+23a^{7}+26a^{6}-29a^{5}-50a^{4}+15a^{3}+23a^{2}-2a-3$ | 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: | \( 1547.38105528 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{1}\cdot 1547.38105528 \cdot 1}{2\cdot\sqrt{4712558875285687}}\cr\approx \mathstrut & 0.145026947441 \end{aligned}\]
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 | ${\href{/padicField/2.13.0.1}{13} }$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.5.0.1}{5} }$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.13.0.1}{13} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.13.0.1}{13} }$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.13.0.1}{13} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(4712558875285687\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |