Normalized defining polynomial
\( x^{14} - 7 x^{13} + 20 x^{12} - 19 x^{11} - 36 x^{10} + 112 x^{9} - 84 x^{8} - 65 x^{7} + 167 x^{6} + \cdots - 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[6, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(120844706781744833\) \(\medspace = 17^{7}\cdot 131^{4}\) | 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: | \(16.60\) | 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: | $17^{1/2}131^{2/3}\approx 106.35028493145434$ | ||
Ramified primes: | \(17\), \(131\) | 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: | \(\Q(\sqrt{17}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}+\frac{2}{5}a^{10}-\frac{2}{5}a^{8}-\frac{2}{5}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{1305715}a^{13}-\frac{66946}{1305715}a^{12}+\frac{345577}{1305715}a^{11}-\frac{531882}{1305715}a^{10}-\frac{65172}{1305715}a^{9}+\frac{30961}{261143}a^{8}+\frac{89138}{261143}a^{7}-\frac{283226}{1305715}a^{6}-\frac{127741}{261143}a^{5}+\frac{464208}{1305715}a^{4}+\frac{308533}{1305715}a^{3}-\frac{396252}{1305715}a^{2}-\frac{104223}{1305715}a+\frac{409307}{1305715}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $\frac{9627}{1305715}a^{13}-\frac{249361}{1305715}a^{12}+\frac{1213674}{1305715}a^{11}-\frac{2286357}{1305715}a^{10}-\frac{667644}{1305715}a^{9}+\frac{9888068}{1305715}a^{8}-\frac{12717867}{1305715}a^{7}-\frac{2634069}{1305715}a^{6}+\frac{20429197}{1305715}a^{5}-\frac{17767467}{1305715}a^{4}+\frac{1051281}{1305715}a^{3}+\frac{9982374}{1305715}a^{2}-\frac{2097827}{261143}a+\frac{2623192}{1305715}$, $\frac{636964}{1305715}a^{13}-\frac{4329762}{1305715}a^{12}+\frac{11813533}{1305715}a^{11}-\frac{9394634}{1305715}a^{10}-\frac{25735113}{1305715}a^{9}+\frac{67340401}{1305715}a^{8}-\frac{40025019}{1305715}a^{7}-\frac{51053488}{1305715}a^{6}+\frac{98688704}{1305715}a^{5}-\frac{54656699}{1305715}a^{4}-\frac{17030848}{1305715}a^{3}+\frac{47895743}{1305715}a^{2}-\frac{5502477}{261143}a+\frac{3537899}{1305715}$, $\frac{122108}{261143}a^{13}-\frac{866268}{261143}a^{12}+\frac{2491519}{261143}a^{11}-\frac{2350014}{261143}a^{10}-\frac{4651368}{261143}a^{9}+\frac{14194607}{261143}a^{8}-\frac{9810114}{261143}a^{7}-\frac{9871780}{261143}a^{6}+\frac{21041679}{261143}a^{5}-\frac{11418265}{261143}a^{4}-\frac{3625619}{261143}a^{3}+\frac{9481544}{261143}a^{2}-\frac{5664268}{261143}a+\frac{806101}{261143}$, $a$, $\frac{341377}{1305715}a^{13}-\frac{2245284}{1305715}a^{12}+\frac{5912139}{1305715}a^{11}-\frac{4254188}{1305715}a^{10}-\frac{13201109}{1305715}a^{9}+\frac{31478164}{1305715}a^{8}-\frac{16316111}{1305715}a^{7}-\frac{23977323}{1305715}a^{6}+\frac{41844801}{1305715}a^{5}-\frac{23758573}{1305715}a^{4}-\frac{5959109}{1305715}a^{3}+\frac{4224830}{261143}a^{2}-\frac{11397328}{1305715}a+\frac{1605588}{1305715}$, $\frac{377049}{1305715}a^{13}-\frac{2190261}{1305715}a^{12}+\frac{4773853}{1305715}a^{11}-\frac{287082}{1305715}a^{10}-\frac{17761138}{1305715}a^{9}+\frac{26590529}{1305715}a^{8}+\frac{926024}{1305715}a^{7}-\frac{7591078}{261143}a^{6}+\frac{36158216}{1305715}a^{5}-\frac{2716987}{1305715}a^{4}-\frac{21510248}{1305715}a^{3}+\frac{19081966}{1305715}a^{2}-\frac{1423859}{1305715}a-\frac{711668}{1305715}$, $\frac{627337}{1305715}a^{13}-\frac{4080401}{1305715}a^{12}+\frac{10599859}{1305715}a^{11}-\frac{7108277}{1305715}a^{10}-\frac{25067469}{1305715}a^{9}+\frac{57452333}{1305715}a^{8}-\frac{27307152}{1305715}a^{7}-\frac{48419419}{1305715}a^{6}+\frac{78259507}{1305715}a^{5}-\frac{36889232}{1305715}a^{4}-\frac{18082129}{1305715}a^{3}+\frac{37913369}{1305715}a^{2}-\frac{3404650}{261143}a+\frac{2220422}{1305715}$, $\frac{182252}{1305715}a^{13}-\frac{1486004}{1305715}a^{12}+\frac{4853524}{1305715}a^{11}-\frac{6282953}{1305715}a^{10}-\frac{4860849}{1305715}a^{9}+\frac{25223154}{1305715}a^{8}-\frac{23355711}{1305715}a^{7}-\frac{9336148}{1305715}a^{6}+\frac{32393751}{1305715}a^{5}-\frac{22641878}{1305715}a^{4}+\frac{139841}{1305715}a^{3}+\frac{2260116}{261143}a^{2}-\frac{9492953}{1305715}a+\frac{2304843}{1305715}$, $\frac{385536}{1305715}a^{13}-\frac{2897224}{1305715}a^{12}+\frac{8967107}{1305715}a^{11}-\frac{10697438}{1305715}a^{10}-\frac{10724167}{1305715}a^{9}+\frac{47501571}{1305715}a^{8}-\frac{45120469}{1305715}a^{7}-\frac{3187425}{261143}a^{6}+\frac{68395864}{1305715}a^{5}-\frac{54293303}{1305715}a^{4}+\frac{3859333}{1305715}a^{3}+\frac{31103374}{1305715}a^{2}-\frac{26020561}{1305715}a+\frac{4836658}{1305715}$ | 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: | \( 1409.11802093 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{4}\cdot 1409.11802093 \cdot 1}{2\cdot\sqrt{120844706781744833}}\cr\approx \mathstrut & 0.202163619571 \end{aligned}\]
Galois group
$C_2\times A_7$ (as 14T47):
A non-solvable group of order 5040 |
The 18 conjugacy class representatives for $A_7\times C_2$ |
Character table for $A_7\times C_2$ |
Intermediate fields
\(\Q(\sqrt{17}) \), 7.3.4959529.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 30 siblings: | data not computed |
Degree 42 siblings: | 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 | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.14.0.1}{14} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.14.0.1}{14} }$ | ${\href{/padicField/31.14.0.1}{14} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.7.0.1}{7} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(17\) | 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.6.3.1 | $x^{6} + 459 x^{5} + 70280 x^{4} + 3597823 x^{3} + 1271380 x^{2} + 4696159 x + 50437479$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(131\) | 131.2.0.1 | $x^{2} + 127 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
131.2.0.1 | $x^{2} + 127 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
131.2.0.1 | $x^{2} + 127 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
131.2.0.1 | $x^{2} + 127 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
131.6.4.1 | $x^{6} + 381 x^{5} + 48393 x^{4} + 2050169 x^{3} + 146697 x^{2} + 6338649 x + 268255520$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |