Normalized defining polynomial
\( x^{14} - x^{13} + x^{12} + x^{11} - 3 x^{10} + 7 x^{9} - 9 x^{8} + x^{7} + 13 x^{6} - 15 x^{5} + \cdots + 1 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-818418501973163\) \(\medspace = -\,7\cdot 13^{4}\cdot 29\cdot 109^{4}\) | 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: | \(11.62\) | 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: | $7^{1/2}13^{1/2}29^{1/2}109^{1/2}\approx 536.3310544803461$ | ||
Ramified primes: | \(7\), \(13\), \(29\), \(109\) | 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: | \(\Q(\sqrt{-203}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{18703}a^{13}-\frac{4387}{18703}a^{12}-\frac{4004}{18703}a^{11}-\frac{572}{18703}a^{10}+\frac{2587}{18703}a^{9}+\frac{6146}{18703}a^{8}-\frac{5342}{18703}a^{7}-\frac{4846}{18703}a^{6}+\frac{7961}{18703}a^{5}+\frac{1540}{18703}a^{4}-\frac{2653}{18703}a^{3}+\frac{2794}{18703}a^{2}-\frac{4021}{18703}a-\frac{823}{18703}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $a$, $\frac{279}{18703}a^{13}-\frac{8278}{18703}a^{12}+\frac{5064}{18703}a^{11}-\frac{9964}{18703}a^{10}-\frac{7644}{18703}a^{9}+\frac{12761}{18703}a^{8}-\frac{50287}{18703}a^{7}+\frac{50691}{18703}a^{6}-\frac{4538}{18703}a^{5}-\frac{75321}{18703}a^{4}+\frac{64042}{18703}a^{3}-\frac{6000}{18703}a^{2}+\frac{321}{18703}a+\frac{13522}{18703}$, $\frac{4979}{18703}a^{13}+\frac{2231}{18703}a^{12}+\frac{1482}{18703}a^{11}+\frac{13571}{18703}a^{10}-\frac{5694}{18703}a^{9}+\frac{21529}{18703}a^{8}-\frac{2152}{18703}a^{7}-\frac{38770}{18703}a^{6}+\frac{62271}{18703}a^{5}-\frac{570}{18703}a^{4}-\frac{42375}{18703}a^{3}+\frac{14997}{18703}a^{2}+\frac{10354}{18703}a-\frac{20463}{18703}$, $\frac{8028}{18703}a^{13}-\frac{1087}{18703}a^{12}+\frac{6345}{18703}a^{11}+\frac{8922}{18703}a^{10}-\frac{10597}{18703}a^{9}+\frac{38980}{18703}a^{8}-\frac{37003}{18703}a^{7}-\frac{20151}{18703}a^{6}+\frac{58866}{18703}a^{5}-\frac{18266}{18703}a^{4}-\frac{14270}{18703}a^{3}-\frac{32071}{18703}a^{2}+\frac{790}{18703}a-\frac{4885}{18703}$, $\frac{6856}{18703}a^{13}-\frac{2848}{18703}a^{12}+\frac{4580}{18703}a^{11}+\frac{5998}{18703}a^{10}-\frac{12675}{18703}a^{9}+\frac{36523}{18703}a^{8}-\frac{41684}{18703}a^{7}-\frac{7648}{18703}a^{6}+\frac{61371}{18703}a^{5}-\frac{27658}{18703}a^{4}-\frac{9652}{18703}a^{3}-\frac{33614}{18703}a^{2}+\frac{18949}{18703}a+\frac{5818}{18703}$, $\frac{4282}{18703}a^{13}-\frac{7322}{18703}a^{12}+\frac{5523}{18703}a^{11}+\frac{789}{18703}a^{10}-\frac{13345}{18703}a^{9}+\frac{39457}{18703}a^{8}-\frac{56784}{18703}a^{7}+\frac{28461}{18703}a^{6}+\frac{49542}{18703}a^{5}-\frac{82691}{18703}a^{4}+\frac{48684}{18703}a^{3}-\frac{24715}{18703}a^{2}+\frac{7541}{18703}a+\frac{10781}{18703}$ | 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: | \( 39.7279937779 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 39.7279937779 \cdot 1}{2\cdot\sqrt{818418501973163}}\cr\approx \mathstrut & 0.268434293360 \end{aligned}\]
Galois group
$C_2^7.\GL(3,2)$ (as 14T51):
A non-solvable group of order 21504 |
The 48 conjugacy class representatives for $C_2^7.\GL(3,2)$ |
Character table for $C_2^7.\GL(3,2)$ |
Intermediate fields
7.3.2007889.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 14 sibling: | data not computed |
Degree 28 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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(29\) | 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
29.6.0.1 | $x^{6} + x^{4} + 25 x^{3} + 17 x^{2} + 13 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
29.6.0.1 | $x^{6} + x^{4} + 25 x^{3} + 17 x^{2} + 13 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(109\) | 109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
109.4.2.1 | $x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
109.4.0.1 | $x^{4} + 11 x^{2} + 98 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
109.4.2.1 | $x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |