Normalized defining polynomial
\( x^{14} - x^{13} + 5 x^{12} - 3 x^{11} + 9 x^{10} - 25 x^{9} + 16 x^{8} - 85 x^{7} + 14 x^{6} + \cdots + 128 \)
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)
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Discriminant: | \(-105496092121152103\) \(\medspace = -\,7^{7}\cdot 71^{6}\) | 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: | \(16.44\) | 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: | $7^{1/2}71^{1/2}\approx 22.293496809607955$ | ||
Ramified primes: | \(7\), \(71\) | 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: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}+\frac{1}{8}a^{8}+\frac{1}{8}a^{7}-\frac{3}{8}a^{6}+\frac{3}{8}a^{5}-\frac{1}{2}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}+\frac{1}{16}a^{9}+\frac{1}{16}a^{8}+\frac{5}{16}a^{7}+\frac{3}{16}a^{6}-\frac{1}{4}a^{5}-\frac{1}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{544}a^{12}+\frac{3}{544}a^{11}-\frac{11}{544}a^{10}+\frac{5}{544}a^{9}-\frac{71}{544}a^{8}-\frac{41}{544}a^{7}+\frac{9}{68}a^{6}+\frac{127}{544}a^{5}-\frac{135}{272}a^{4}-\frac{15}{68}a^{3}+\frac{1}{68}a^{2}+\frac{9}{34}a+\frac{5}{17}$, $\frac{1}{760545728}a^{13}-\frac{339987}{760545728}a^{12}-\frac{14926593}{760545728}a^{11}-\frac{12178973}{760545728}a^{10}+\frac{14923871}{760545728}a^{9}-\frac{35430315}{760545728}a^{8}+\frac{5509633}{380272864}a^{7}+\frac{71658635}{760545728}a^{6}+\frac{188111}{505682}a^{5}+\frac{2807973}{47534108}a^{4}-\frac{29316207}{95068216}a^{3}+\frac{11880073}{47534108}a^{2}+\frac{4163979}{23767054}a+\frac{595858}{11883527}$
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: | $\frac{1240969}{760545728}a^{13}+\frac{3609391}{760545728}a^{12}-\frac{2661863}{760545728}a^{11}+\frac{28296473}{760545728}a^{10}-\frac{14314707}{760545728}a^{9}+\frac{1262219}{44737984}a^{8}-\frac{18509997}{190136432}a^{7}+\frac{65159279}{760545728}a^{6}-\frac{3781563}{8090912}a^{5}+\frac{127023}{1398062}a^{4}-\frac{2073043}{11883527}a^{3}+\frac{17626941}{47534108}a^{2}+\frac{1191458}{11883527}a+\frac{11207819}{11883527}$, $\frac{40389}{23767054}a^{13}+\frac{30021}{23767054}a^{12}+\frac{16935}{11883527}a^{11}+\frac{389707}{23767054}a^{10}-\frac{224861}{23767054}a^{9}+\frac{227484}{11883527}a^{8}-\frac{71975}{1398062}a^{7}+\frac{444877}{11883527}a^{6}-\frac{34734}{252841}a^{5}+\frac{660332}{11883527}a^{4}-\frac{2889064}{11883527}a^{3}+\frac{2420752}{11883527}a^{2}+\frac{603645}{23767054}a-\frac{4176192}{11883527}$, $\frac{396719}{760545728}a^{13}-\frac{4067327}{760545728}a^{12}+\frac{8902631}{760545728}a^{11}-\frac{18591617}{760545728}a^{10}+\frac{29677851}{760545728}a^{9}-\frac{11245459}{760545728}a^{8}+\frac{1783617}{11184496}a^{7}-\frac{60398735}{760545728}a^{6}+\frac{3325275}{8090912}a^{5}-\frac{7819363}{23767054}a^{4}+\frac{10889603}{95068216}a^{3}-\frac{32238759}{23767054}a^{2}-\frac{12085927}{23767054}a-\frac{17803019}{11883527}$, $\frac{405017}{190136432}a^{13}-\frac{4485519}{380272864}a^{12}+\frac{12382335}{380272864}a^{11}-\frac{24120027}{380272864}a^{10}+\frac{30788813}{380272864}a^{9}-\frac{43160411}{380272864}a^{8}+\frac{66557633}{380272864}a^{7}-\frac{62173407}{190136432}a^{6}+\frac{4155751}{8090912}a^{5}-\frac{95385483}{190136432}a^{4}+\frac{40049919}{47534108}a^{3}-\frac{19209211}{23767054}a^{2}+\frac{8572458}{11883527}a-\frac{13157850}{11883527}$, $\frac{479913}{380272864}a^{13}-\frac{1957359}{380272864}a^{12}-\frac{455461}{380272864}a^{11}+\frac{2372811}{380272864}a^{10}-\frac{662489}{22368992}a^{9}+\frac{24117061}{380272864}a^{8}-\frac{4566423}{190136432}a^{7}+\frac{82237487}{380272864}a^{6}-\frac{4819}{59492}a^{5}+\frac{11755479}{23767054}a^{4}-\frac{102118419}{95068216}a^{3}+\frac{6773309}{23767054}a^{2}-\frac{43095117}{23767054}a-\frac{3396438}{11883527}$, $\frac{2194113}{380272864}a^{13}-\frac{11547439}{380272864}a^{12}+\frac{28935925}{380272864}a^{11}-\frac{68909803}{380272864}a^{10}+\frac{80258189}{380272864}a^{9}-\frac{162044573}{380272864}a^{8}+\frac{7064073}{11883527}a^{7}-\frac{429597639}{380272864}a^{6}+\frac{4247275}{2022728}a^{5}-\frac{351344589}{190136432}a^{4}+\frac{321200207}{95068216}a^{3}-\frac{71144593}{47534108}a^{2}+\frac{19926711}{11883527}a-\frac{16217824}{11883527}$ | 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: | \( 451.410051802 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 451.410051802 \cdot 1}{2\cdot\sqrt{105496092121152103}}\cr\approx \mathstrut & 0.268647078579 \end{aligned}\]
Galois group
A solvable group of order 28 |
The 10 conjugacy class representatives for $D_{14}$ |
Character table for $D_{14}$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), 7.1.357911.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 28 |
Degree 14 sibling: | 14.2.7490222540601799313.1 |
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 | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.14.0.1}{14} }$ | R | ${\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.2.0.1}{2} }^{7}$ | ${\href{/padicField/19.14.0.1}{14} }$ | ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{7}$ | ${\href{/padicField/37.7.0.1}{7} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{7}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{7}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(71\) | $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.71.2t1.a.a | $1$ | $ 71 $ | \(\Q(\sqrt{-71}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.497.2t1.a.a | $1$ | $ 7 \cdot 71 $ | \(\Q(\sqrt{497}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 2.71.7t2.a.c | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.71.7t2.a.a | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.3479.14t3.b.c | $2$ | $ 7^{2} \cdot 71 $ | 14.0.105496092121152103.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
* | 2.71.7t2.a.b | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
* | 2.3479.14t3.b.b | $2$ | $ 7^{2} \cdot 71 $ | 14.0.105496092121152103.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
* | 2.3479.14t3.b.a | $2$ | $ 7^{2} \cdot 71 $ | 14.0.105496092121152103.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |