Normalized defining polynomial
\( x^{13} - 2 x^{12} + x^{11} + 5 x^{10} - 10 x^{9} + 4 x^{8} + 8 x^{7} - 13 x^{6} + 5 x^{5} + 10 x^{4} + \cdots + 1 \)
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: | \(-215762905145828\) \(\medspace = -\,2^{2}\cdot 953\cdot 10313\cdot 5488313\) | 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: | \(12.67\) | 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: | $2\cdot 953^{1/2}10313^{1/2}5488313^{1/2}\approx 14688870.111272275$ | ||
Ramified primes: | \(2\), \(953\), \(10313\), \(5488313\) | 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: | $\Q(\sqrt{-53940726286457}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{373}a^{12}+\frac{111}{373}a^{11}-\frac{138}{373}a^{10}+\frac{77}{373}a^{9}+\frac{112}{373}a^{8}-\frac{22}{373}a^{7}+\frac{133}{373}a^{6}+\frac{96}{373}a^{5}+\frac{36}{373}a^{4}-\frac{25}{373}a^{3}+\frac{150}{373}a^{2}+\frac{162}{373}a+\frac{33}{373}$
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: | $\frac{159}{373}a^{12}-\frac{255}{373}a^{11}+\frac{65}{373}a^{10}+\frac{680}{373}a^{9}-\frac{1215}{373}a^{8}+\frac{232}{373}a^{7}+\frac{632}{373}a^{6}-\frac{1148}{373}a^{5}+\frac{502}{373}a^{4}+\frac{874}{373}a^{3}-\frac{22}{373}a^{2}-\frac{725}{373}a-\frac{348}{373}$, $\frac{289}{373}a^{12}-\frac{745}{373}a^{11}+\frac{775}{373}a^{10}+\frac{992}{373}a^{9}-\frac{3440}{373}a^{8}+\frac{3340}{373}a^{7}+\frac{391}{373}a^{6}-\frac{3961}{373}a^{5}+\frac{3690}{373}a^{4}+\frac{608}{373}a^{3}-\frac{2902}{373}a^{2}+\frac{939}{373}a+\frac{585}{373}$, $\frac{188}{373}a^{12}-\frac{393}{373}a^{11}+\frac{539}{373}a^{10}+\frac{302}{373}a^{9}-\frac{1324}{373}a^{8}+\frac{1832}{373}a^{7}-\frac{1106}{373}a^{6}-\frac{229}{373}a^{5}+\frac{800}{373}a^{4}-\frac{224}{373}a^{3}-\frac{148}{373}a^{2}+\frac{243}{373}a-\frac{137}{373}$, $\frac{471}{373}a^{12}-\frac{1058}{373}a^{11}+\frac{1023}{373}a^{10}+\frac{1578}{373}a^{9}-\frac{4690}{373}a^{8}+\frac{4185}{373}a^{7}+\frac{352}{373}a^{6}-\frac{4766}{373}a^{5}+\frac{4274}{373}a^{4}+\frac{1280}{373}a^{3}-\frac{3204}{373}a^{2}+\frac{583}{373}a+\frac{623}{373}$, $\frac{352}{373}a^{12}-\frac{839}{373}a^{11}+\frac{660}{373}a^{10}+\frac{1367}{373}a^{9}-\frac{3844}{373}a^{8}+\frac{2700}{373}a^{7}+\frac{1310}{373}a^{6}-\frac{4254}{373}a^{5}+\frac{2974}{373}a^{4}+\frac{2390}{373}a^{3}-\frac{3523}{373}a^{2}-\frac{45}{373}a+\frac{799}{373}$, $\frac{443}{373}a^{12}-\frac{1182}{373}a^{11}+\frac{1157}{373}a^{10}+\frac{1660}{373}a^{9}-\frac{5588}{373}a^{8}+\frac{5174}{373}a^{7}+\frac{1104}{373}a^{6}-\frac{6708}{373}a^{5}+\frac{6250}{373}a^{4}+\frac{1234}{373}a^{3}-\frac{5166}{373}a^{2}+\frac{1269}{373}a+\frac{1564}{373}$, $\frac{3}{373}a^{12}-\frac{40}{373}a^{11}-\frac{41}{373}a^{10}+\frac{231}{373}a^{9}-\frac{410}{373}a^{8}-\frac{66}{373}a^{7}+\frac{1145}{373}a^{6}-\frac{1204}{373}a^{5}+\frac{108}{373}a^{4}+\frac{1417}{373}a^{3}-\frac{1415}{373}a^{2}-\frac{260}{373}a+\frac{845}{373}$ | 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: | \( 143.677539316 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{5}\cdot 143.677539316 \cdot 1}{2\cdot\sqrt{215762905145828}}\cr\approx \mathstrut & 0.383142054819 \end{aligned}\]
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 | ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.5.0.1}{5} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.11.0.1 | $x^{11} + x^{2} + 1$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
\(953\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(10313\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(5488313\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |