Normalized defining polynomial
\( x^{24} - 81x^{20} + 2913x^{16} - 59920x^{12} + 745728x^{8} - 5308416x^{4} + 16777216 \)
Invariants
Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(5491974027251682803463706102741604499456\) \(\medspace = 2^{48}\cdot 3^{28}\cdot 31^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(45.27\) | 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: | $2^{2}3^{7/6}31^{1/2}\approx 80.23840777241452$ | ||
Ramified primes: | \(2\), \(3\), \(31\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{2048}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{7}+\frac{1}{8}a^{3}$, $\frac{1}{80}a^{12}-\frac{33}{80}a^{8}+\frac{17}{80}a^{4}+\frac{1}{5}$, $\frac{1}{160}a^{13}-\frac{33}{160}a^{9}+\frac{17}{160}a^{5}-\frac{2}{5}a$, $\frac{1}{320}a^{14}-\frac{33}{320}a^{10}-\frac{143}{320}a^{6}+\frac{3}{10}a^{2}$, $\frac{1}{640}a^{15}-\frac{33}{640}a^{11}+\frac{177}{640}a^{7}-\frac{7}{20}a^{3}$, $\frac{1}{1280}a^{16}-\frac{1}{1280}a^{12}-\frac{239}{1280}a^{8}+\frac{1}{4}a^{4}+\frac{2}{5}$, $\frac{1}{5120}a^{17}+\frac{3}{1024}a^{13}-\frac{767}{5120}a^{9}+\frac{37}{320}a^{5}+\frac{3}{20}a$, $\frac{1}{20480}a^{18}-\frac{17}{20480}a^{14}-\frac{2271}{20480}a^{10}+\frac{483}{1280}a^{6}-\frac{19}{80}a^{2}$, $\frac{1}{81920}a^{19}+\frac{47}{81920}a^{15}+\frac{737}{81920}a^{11}-\frac{409}{5120}a^{7}-\frac{27}{64}a^{3}$, $\frac{1}{4259840}a^{20}+\frac{943}{4259840}a^{16}+\frac{10081}{4259840}a^{12}+\frac{62367}{266240}a^{8}+\frac{77}{3328}a^{4}-\frac{23}{65}$, $\frac{1}{17039360}a^{21}+\frac{943}{17039360}a^{17}+\frac{10081}{17039360}a^{13}+\frac{62367}{1064960}a^{9}+\frac{3405}{13312}a^{5}+\frac{107}{260}a$, $\frac{1}{68157440}a^{22}+\frac{943}{68157440}a^{18}-\frac{19283}{13631488}a^{14}+\frac{56403}{851968}a^{10}+\frac{9953}{266240}a^{6}-\frac{309}{1040}a^{2}$, $\frac{1}{272629760}a^{23}+\frac{943}{272629760}a^{19}+\frac{116577}{272629760}a^{15}-\frac{157281}{17039360}a^{11}+\frac{157217}{1064960}a^{7}-\frac{2077}{4160}a^{3}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $5$ |
Class group and class number
$C_{2}\times C_{6}\times C_{24}$, which has order $288$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{183}{4259840} a^{21} - \frac{12967}{4259840} a^{17} + \frac{392983}{4259840} a^{13} - \frac{409507}{266240} a^{9} + \frac{237323}{16640} a^{5} - \frac{15263}{260} a \) (order $24$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{1007}{68157440}a^{22}-\frac{62111}{68157440}a^{18}+\frac{337027}{13631488}a^{14}-\frac{1576591}{4259840}a^{10}+\frac{806607}{266240}a^{6}-\frac{10759}{1040}a^{2}+1$, $\frac{341}{272629760}a^{23}-\frac{147}{5242880}a^{22}-\frac{21221}{272629760}a^{19}+\frac{9091}{5242880}a^{18}+\frac{638773}{272629760}a^{15}-\frac{249267}{5242880}a^{14}-\frac{679141}{17039360}a^{11}+\frac{237507}{327680}a^{10}+\frac{409381}{1064960}a^{7}-\frac{124611}{20480}a^{6}-\frac{1717}{1040}a^{3}+\frac{219}{10}a^{2}$, $\frac{477}{17039360}a^{21}-\frac{32749}{17039360}a^{17}+\frac{978109}{17039360}a^{13}-\frac{1005197}{1064960}a^{9}+\frac{575309}{66560}a^{5}-\frac{2291}{65}a-1$, $\frac{457}{68157440}a^{22}-\frac{8325}{13631488}a^{18}+\frac{1458729}{68157440}a^{14}-\frac{339893}{851968}a^{10}+\frac{1091977}{266240}a^{6}-\frac{19247}{1040}a^{2}$, $\frac{359}{27262976}a^{23}-\frac{361}{13631488}a^{22}-\frac{231}{4259840}a^{21}+\frac{101}{4259840}a^{20}-\frac{97779}{136314880}a^{19}+\frac{124957}{68157440}a^{18}+\frac{11799}{4259840}a^{17}-\frac{11253}{4259840}a^{16}+\frac{484103}{27262976}a^{15}-\frac{3782637}{68157440}a^{14}-\frac{268679}{4259840}a^{13}+\frac{432453}{4259840}a^{12}-\frac{2067267}{8519680}a^{11}+\frac{3955197}{4259840}a^{10}+\frac{204183}{266240}a^{9}-\frac{533317}{266240}a^{8}+\frac{937347}{532480}a^{7}-\frac{461593}{53248}a^{6}-\frac{74167}{16640}a^{5}+\frac{358997}{16640}a^{4}-\frac{10397}{2080}a^{3}+\frac{18957}{520}a^{2}+\frac{707}{130}a-\frac{6834}{65}$, $\frac{211}{68157440}a^{23}-\frac{113}{8519680}a^{22}+\frac{209}{8519680}a^{21}+\frac{71}{851968}a^{20}-\frac{19843}{68157440}a^{19}+\frac{7009}{8519680}a^{18}-\frac{3181}{1703936}a^{17}-\frac{24659}{4259840}a^{16}+\frac{681907}{68157440}a^{15}-\frac{194417}{8519680}a^{14}+\frac{509489}{8519680}a^{13}+\frac{743299}{4259840}a^{12}-\frac{773599}{4259840}a^{11}+\frac{37775}{106496}a^{10}-\frac{550193}{532480}a^{9}-\frac{778819}{266240}a^{8}+\frac{97523}{53248}a^{7}-\frac{101667}{33280}a^{6}+\frac{326321}{33280}a^{5}+\frac{91399}{3328}a^{4}-\frac{6875}{832}a^{3}+\frac{12017}{1040}a^{2}-\frac{2644}{65}a-\frac{7463}{65}$, $\frac{101}{8519680}a^{21}-\frac{101}{4259840}a^{20}-\frac{11253}{8519680}a^{17}+\frac{11253}{4259840}a^{16}+\frac{432453}{8519680}a^{13}-\frac{432453}{4259840}a^{12}-\frac{533317}{532480}a^{9}+\frac{533317}{266240}a^{8}+\frac{358997}{33280}a^{5}-\frac{358997}{16640}a^{4}-\frac{3417}{65}a+\frac{6834}{65}$, $\frac{359}{27262976}a^{23}+\frac{361}{13631488}a^{22}-\frac{231}{4259840}a^{21}-\frac{101}{4259840}a^{20}-\frac{97779}{136314880}a^{19}-\frac{124957}{68157440}a^{18}+\frac{11799}{4259840}a^{17}+\frac{11253}{4259840}a^{16}+\frac{484103}{27262976}a^{15}+\frac{3782637}{68157440}a^{14}-\frac{268679}{4259840}a^{13}-\frac{432453}{4259840}a^{12}-\frac{2067267}{8519680}a^{11}-\frac{3955197}{4259840}a^{10}+\frac{204183}{266240}a^{9}+\frac{533317}{266240}a^{8}+\frac{937347}{532480}a^{7}+\frac{461593}{53248}a^{6}-\frac{74167}{16640}a^{5}-\frac{358997}{16640}a^{4}-\frac{10397}{2080}a^{3}-\frac{18957}{520}a^{2}+\frac{707}{130}a+\frac{6834}{65}$, $\frac{1125}{54525952}a^{23}+\frac{69}{17039360}a^{22}+\frac{41}{655360}a^{21}+\frac{59}{851968}a^{20}-\frac{349897}{272629760}a^{19}-\frac{3989}{17039360}a^{18}-\frac{485}{131072}a^{17}-\frac{18007}{4259840}a^{16}+\frac{9591129}{272629760}a^{15}+\frac{112357}{17039360}a^{14}+\frac{63241}{655360}a^{13}+\frac{501191}{4259840}a^{12}-\frac{9136073}{17039360}a^{11}-\frac{108929}{1064960}a^{10}-\frac{56977}{40960}a^{9}-\frac{486679}{266240}a^{8}+\frac{4822409}{1064960}a^{7}+\frac{55009}{66560}a^{6}+\frac{28049}{2560}a^{5}+\frac{258343}{16640}a^{4}-\frac{17163}{1040}a^{3}-\frac{2643}{1040}a^{2}-\frac{719}{20}a-\frac{720}{13}$, $\frac{7573}{272629760}a^{23}+\frac{111}{17039360}a^{22}+\frac{111}{8519680}a^{21}+\frac{47}{212992}a^{20}-\frac{493093}{272629760}a^{19}-\frac{1363}{3407872}a^{18}-\frac{1363}{1703936}a^{17}-\frac{3103}{212992}a^{16}+\frac{14116469}{272629760}a^{15}+\frac{192143}{17039360}a^{14}+\frac{192143}{8519680}a^{13}+\frac{449611}{1064960}a^{12}-\frac{13973333}{17039360}a^{11}-\frac{186183}{1064960}a^{10}-\frac{186183}{532480}a^{9}-\frac{450223}{66560}a^{8}+\frac{7653013}{1064960}a^{7}+\frac{100663}{66560}a^{6}+\frac{100663}{33280}a^{5}+\frac{249907}{4160}a^{4}-\frac{113599}{4160}a^{3}-\frac{561}{104}a^{2}-\frac{561}{52}a-\frac{15146}{65}$, $\frac{3}{327680}a^{23}-\frac{55}{6815744}a^{22}-\frac{51}{3407872}a^{21}-\frac{209}{4259840}a^{20}-\frac{183}{327680}a^{19}+\frac{10243}{34078720}a^{18}+\frac{19119}{17039360}a^{17}+\frac{3181}{851968}a^{16}+\frac{4903}{327680}a^{15}-\frac{113203}{34078720}a^{14}-\frac{593823}{17039360}a^{13}-\frac{509489}{4259840}a^{12}-\frac{3577}{16384}a^{11}-\frac{61437}{2129920}a^{10}+\frac{632831}{1064960}a^{9}+\frac{550193}{266240}a^{8}+\frac{8749}{5120}a^{7}+\frac{28537}{26624}a^{6}-\frac{373983}{66560}a^{5}-\frac{326321}{16640}a^{4}-\frac{349}{64}a^{3}-\frac{1061}{130}a^{2}+\frac{6099}{260}a+\frac{5288}{65}$ (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: | \( 1243512309.688251 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 1243512309.688251 \cdot 288}{24\cdot\sqrt{5491974027251682803463706102741604499456}}\cr\approx \mathstrut & 0.762299294370137 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2\times D_6$ (as 24T30):
A solvable group of order 48 |
The 24 conjugacy class representatives for $C_2^2\times D_6$ |
Character table for $C_2^2\times D_6$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 24 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 | R | ${\href{/padicField/5.2.0.1}{2} }^{12}$ | ${\href{/padicField/7.6.0.1}{6} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{12}$ | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{4}$ | R | ${\href{/padicField/37.2.0.1}{2} }^{12}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.2.0.1}{2} }^{12}$ | ${\href{/padicField/47.2.0.1}{2} }^{12}$ | ${\href{/padicField/53.2.0.1}{2} }^{12}$ | ${\href{/padicField/59.2.0.1}{2} }^{12}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
\(3\) | 3.12.14.11 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 792 x^{8} + 1728 x^{7} + 2918 x^{6} + 3684 x^{5} + 3156 x^{4} + 1376 x^{3} - 36 x^{2} - 168 x + 25$ | $6$ | $2$ | $14$ | $D_6$ | $[3/2]_{2}^{2}$ |
3.12.14.11 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 792 x^{8} + 1728 x^{7} + 2918 x^{6} + 3684 x^{5} + 3156 x^{4} + 1376 x^{3} - 36 x^{2} - 168 x + 25$ | $6$ | $2$ | $14$ | $D_6$ | $[3/2]_{2}^{2}$ | |
\(31\) | 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |