Normalized defining polynomial
\( x^{14} - 56 x^{11} + 196 x^{10} - 224 x^{9} + 840 x^{8} - 5648 x^{7} + 15876 x^{6} - 23520 x^{5} + \cdots - 6272 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(36202216328934500847321088\) \(\medspace = 2^{27}\cdot 7^{12}\cdot 11^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(66.93\) | 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))
| |
Galois root discriminant: | $2^{2}7^{47/42}11^{1/2}\approx 117.0742228154606$ | ||
Ramified primes: | \(2\), \(7\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{22}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{24}a^{7}-\frac{1}{6}a^{4}+\frac{1}{12}a^{3}+\frac{1}{3}a^{2}+\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{48}a^{8}-\frac{1}{12}a^{5}+\frac{1}{24}a^{4}+\frac{1}{6}a^{3}-\frac{5}{12}a^{2}+\frac{1}{3}a$, $\frac{1}{96}a^{9}-\frac{1}{24}a^{6}+\frac{1}{48}a^{5}+\frac{1}{12}a^{4}-\frac{5}{24}a^{3}-\frac{1}{3}a^{2}-\frac{1}{2}a$, $\frac{1}{96}a^{10}+\frac{1}{48}a^{6}+\frac{1}{12}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{6}a^{2}+\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{96}a^{11}-\frac{1}{48}a^{7}+\frac{1}{12}a^{6}+\frac{1}{8}a^{5}-\frac{1}{12}a^{4}-\frac{1}{4}a^{3}-\frac{1}{6}a^{2}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{192}a^{12}-\frac{1}{16}a^{6}+\frac{1}{6}a^{5}+\frac{1}{16}a^{4}+\frac{5}{12}a^{3}+\frac{11}{24}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{38361535296}a^{13}-\frac{9597773}{5480219328}a^{12}-\frac{408291}{456684944}a^{11}-\frac{121511}{171256854}a^{10}+\frac{3324325}{2740109664}a^{9}+\frac{1808257}{456684944}a^{8}+\frac{1052777}{1370054832}a^{7}+\frac{937482331}{9590383824}a^{6}-\frac{72623155}{685027416}a^{5}+\frac{232845463}{1370054832}a^{4}-\frac{14240883}{114171236}a^{3}+\frac{77988275}{685027416}a^{2}-\frac{52038151}{171256854}a-\frac{3120328}{85628427}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | $\frac{612076163}{12787178432}a^{13}+\frac{42663727}{2740109664}a^{12}-\frac{272466287}{2740109664}a^{11}-\frac{8245730027}{2740109664}a^{10}+\frac{10800577175}{1370054832}a^{9}-\frac{3945548519}{1370054832}a^{8}+\frac{24204743909}{685027416}a^{7}-\frac{840136695029}{3196794608}a^{6}+\frac{805506803033}{1370054832}a^{5}-\frac{132019478245}{228342472}a^{4}+\frac{115398861409}{228342472}a^{3}-\frac{104470194113}{171256854}a^{2}+\frac{9050914628}{28542809}a+\frac{20594968069}{85628427}$, $\frac{2053039}{1826739776}a^{13}-\frac{4852417}{1826739776}a^{12}-\frac{7584577}{2740109664}a^{11}-\frac{69804923}{2740109664}a^{10}+\frac{135329627}{685027416}a^{9}-\frac{435815777}{1370054832}a^{8}-\frac{68622713}{685027416}a^{7}+\frac{176856679}{685027416}a^{6}+\frac{1771267325}{1370054832}a^{5}-\frac{2766464021}{1370054832}a^{4}+\frac{506675869}{685027416}a^{3}-\frac{1011447787}{685027416}a^{2}+\frac{275176303}{85628427}a+\frac{137906618}{85628427}$, $\frac{239873303}{38361535296}a^{13}+\frac{3220457}{913369888}a^{12}-\frac{16392413}{2740109664}a^{11}-\frac{478236109}{1370054832}a^{10}+\frac{458007097}{456684944}a^{9}-\frac{394910411}{685027416}a^{8}+\frac{691448861}{171256854}a^{7}-\frac{24941344363}{799198652}a^{6}+\frac{106181837411}{1370054832}a^{5}-\frac{65645778047}{685027416}a^{4}+\frac{60942894785}{685027416}a^{3}-\frac{12212761157}{114171236}a^{2}+\frac{7187706479}{85628427}a-\frac{22976719}{85628427}$, $\frac{11493001}{19180767648}a^{13}-\frac{9270383}{1370054832}a^{12}+\frac{43624315}{2740109664}a^{11}-\frac{128028895}{2740109664}a^{10}+\frac{493562375}{1370054832}a^{9}-\frac{2057449837}{1370054832}a^{8}+\frac{4907807131}{1370054832}a^{7}-\frac{76863516337}{9590383824}a^{6}+\frac{5406940463}{228342472}a^{5}-\frac{6422015377}{114171236}a^{4}+\frac{6717728339}{85628427}a^{3}-\frac{17187481031}{342513708}a^{2}-\frac{20510932}{28542809}a+\frac{1216373399}{85628427}$, $\frac{10147693}{5480219328}a^{13}-\frac{262970837}{5480219328}a^{12}+\frac{37423113}{456684944}a^{11}-\frac{3380742}{28542809}a^{10}+\frac{79889563}{57085618}a^{9}-\frac{5032424905}{685027416}a^{8}+\frac{20624940527}{1370054832}a^{7}-\frac{20052841993}{1370054832}a^{6}+\frac{12292680883}{1370054832}a^{5}-\frac{18921392863}{1370054832}a^{4}+\frac{2578847359}{228342472}a^{3}+\frac{3237569749}{685027416}a^{2}+\frac{25415900}{28542809}a-\frac{347967614}{28542809}$, $\frac{643566989}{6393589216}a^{13}-\frac{67056811}{1370054832}a^{12}+\frac{106543403}{2740109664}a^{11}-\frac{5071515025}{913369888}a^{10}+\frac{15457041509}{685027416}a^{9}-\frac{2880057989}{85628427}a^{8}+\frac{133223996675}{1370054832}a^{7}-\frac{5867874971837}{9590383824}a^{6}+\frac{323156178991}{171256854}a^{5}-\frac{2194668132529}{685027416}a^{4}+\frac{410226251337}{114171236}a^{3}-\frac{629598517013}{171256854}a^{2}+\frac{97046945582}{28542809}a-\frac{153807022477}{85628427}$, $\frac{4846034871}{3196794608}a^{13}+\frac{7996531289}{2740109664}a^{12}+\frac{11230118075}{2740109664}a^{11}-\frac{27647797943}{342513708}a^{10}+\frac{185741700511}{1370054832}a^{9}-\frac{634245077}{456684944}a^{8}+\frac{1607690869567}{1370054832}a^{7}-\frac{10149460159045}{1598397304}a^{6}+\frac{608020509569}{57085618}a^{5}-\frac{1071863070515}{114171236}a^{4}+\frac{949103399791}{114171236}a^{3}-\frac{676206625997}{57085618}a^{2}+\frac{1607656763}{28542809}a+\frac{211347658273}{85628427}$ (assuming GRH) | 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: | \( 650123158.139 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 650123158.139 \cdot 1}{2\cdot\sqrt{36202216328934500847321088}}\cr\approx \mathstrut & 13.2964976943 \end{aligned}\] (assuming GRH)
Galois group
$C_7^2:D_6$ (as 14T25):
A solvable group of order 588 |
The 19 conjugacy class representatives for $C_7^2:D_6$ |
Character table for $C_7^2:D_6$ |
Intermediate fields
\(\Q(\sqrt{22}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 21 siblings: | data not computed |
Degree 28 sibling: | 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 | R | ${\href{/padicField/3.3.0.1}{3} }^{4}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }$ | R | R | ${\href{/padicField/13.3.0.1}{3} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.14.0.1}{14} }$ | ${\href{/padicField/19.14.0.1}{14} }$ | ${\href{/padicField/23.14.0.1}{14} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.14.0.1}{14} }$ | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.14.0.1}{14} }$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.14.0.1}{14} }$ | ${\href{/padicField/53.14.0.1}{14} }$ | ${\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.4.8.3 | $x^{4} + 6 x^{2} + 4 x + 14$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
2.4.8.3 | $x^{4} + 6 x^{2} + 4 x + 14$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
2.4.8.3 | $x^{4} + 6 x^{2} + 4 x + 14$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.6.5.6 | $x^{6} + 28$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
7.7.7.6 | $x^{7} + 28 x + 7$ | $7$ | $1$ | $7$ | $F_7$ | $[7/6]_{6}$ | |
\(11\) | 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |