Normalized defining polynomial
\( x^{16} - 22x^{14} + 89x^{12} - 361x^{10} + 4554x^{8} - 9692x^{6} + 33196x^{4} - 60065x^{2} + 25 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(43149301773875176172265625\) \(\medspace = 5^{8}\cdot 101^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(40.01\) | 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: | $5^{1/2}101^{3/4}\approx 71.24034803863643$ | ||
Ramified primes: | \(5\), \(101\) | 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 not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{10}a^{10}-\frac{1}{2}a^{7}+\frac{3}{10}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}+\frac{1}{10}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{10}a^{11}+\frac{3}{10}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{10}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{10}a^{12}-\frac{1}{5}a^{8}+\frac{1}{10}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{50}a^{13}-\frac{1}{50}a^{11}+\frac{3}{50}a^{9}+\frac{11}{25}a^{7}-\frac{1}{2}a^{6}-\frac{2}{25}a^{5}+\frac{9}{50}a^{3}+\frac{2}{5}a-\frac{1}{2}$, $\frac{1}{25\!\cdots\!50}a^{14}-\frac{468101337070361}{25\!\cdots\!50}a^{12}-\frac{12\!\cdots\!87}{25\!\cdots\!50}a^{10}+\frac{698937158958821}{12\!\cdots\!25}a^{8}-\frac{1}{2}a^{7}+\frac{33\!\cdots\!88}{12\!\cdots\!25}a^{6}-\frac{883021918011041}{23\!\cdots\!50}a^{4}-\frac{665470307877967}{25\!\cdots\!65}a^{2}-\frac{1}{2}a+\frac{19619138519617}{509271609335533}$, $\frac{1}{25\!\cdots\!50}a^{15}+\frac{20585136132586}{12\!\cdots\!25}a^{13}+\frac{154315981456189}{50\!\cdots\!30}a^{11}+\frac{29\!\cdots\!41}{25\!\cdots\!50}a^{9}-\frac{851276323753}{25\!\cdots\!50}a^{7}-\frac{10\!\cdots\!53}{23\!\cdots\!50}a^{5}-\frac{1}{2}a^{4}+\frac{237549725958896}{12\!\cdots\!25}a^{3}-\frac{313080224139363}{50\!\cdots\!30}a-\frac{1}{2}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $5$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: | $\frac{1837169}{170086036115}a^{14}-\frac{48736329}{170086036115}a^{12}+\frac{334801303}{170086036115}a^{10}-\frac{1135669947}{170086036115}a^{8}+\frac{10319493572}{170086036115}a^{6}-\frac{54341924904}{170086036115}a^{4}+\frac{72688861086}{170086036115}a^{2}-\frac{21029840625}{34017207223}$, $\frac{14152966011076}{12\!\cdots\!25}a^{15}+\frac{115361694083}{25\!\cdots\!65}a^{14}-\frac{623298790214571}{25\!\cdots\!50}a^{13}-\frac{4887329918143}{50\!\cdots\!30}a^{12}+\frac{506178075687571}{50\!\cdots\!30}a^{11}+\frac{16559127680761}{50\!\cdots\!30}a^{10}-\frac{10\!\cdots\!13}{25\!\cdots\!50}a^{9}-\frac{35109537920597}{25\!\cdots\!65}a^{8}+\frac{12\!\cdots\!89}{25\!\cdots\!50}a^{7}+\frac{98576962372380}{509271609335533}a^{6}-\frac{25\!\cdots\!21}{23\!\cdots\!50}a^{5}-\frac{120602677896943}{462974190305030}a^{4}+\frac{94\!\cdots\!99}{25\!\cdots\!50}a^{3}+\frac{72\!\cdots\!43}{50\!\cdots\!30}a^{2}-\frac{34\!\cdots\!61}{50\!\cdots\!30}a-\frac{12\!\cdots\!47}{10\!\cdots\!66}$, $\frac{14152966011076}{12\!\cdots\!25}a^{15}-\frac{115361694083}{25\!\cdots\!65}a^{14}-\frac{623298790214571}{25\!\cdots\!50}a^{13}+\frac{4887329918143}{50\!\cdots\!30}a^{12}+\frac{506178075687571}{50\!\cdots\!30}a^{11}-\frac{16559127680761}{50\!\cdots\!30}a^{10}-\frac{10\!\cdots\!13}{25\!\cdots\!50}a^{9}+\frac{35109537920597}{25\!\cdots\!65}a^{8}+\frac{12\!\cdots\!89}{25\!\cdots\!50}a^{7}-\frac{98576962372380}{509271609335533}a^{6}-\frac{25\!\cdots\!21}{23\!\cdots\!50}a^{5}+\frac{120602677896943}{462974190305030}a^{4}+\frac{94\!\cdots\!99}{25\!\cdots\!50}a^{3}-\frac{72\!\cdots\!43}{50\!\cdots\!30}a^{2}-\frac{34\!\cdots\!61}{50\!\cdots\!30}a+\frac{12\!\cdots\!47}{10\!\cdots\!66}$, $\frac{1845712102022}{12\!\cdots\!25}a^{15}+\frac{46441822707}{25\!\cdots\!65}a^{14}-\frac{82402355851071}{25\!\cdots\!50}a^{13}-\frac{1728662320171}{50\!\cdots\!30}a^{12}+\frac{351390238554749}{25\!\cdots\!50}a^{11}+\frac{1902664928017}{50\!\cdots\!30}a^{10}-\frac{686436054152619}{12\!\cdots\!25}a^{9}-\frac{19370051414333}{50\!\cdots\!30}a^{8}+\frac{341352959440403}{509271609335533}a^{7}+\frac{191252737563117}{25\!\cdots\!65}a^{6}-\frac{18\!\cdots\!68}{11\!\cdots\!75}a^{5}+\frac{32588868151679}{462974190305030}a^{4}+\frac{12\!\cdots\!27}{25\!\cdots\!50}a^{3}+\frac{291505556370565}{509271609335533}a^{2}-\frac{51\!\cdots\!13}{50\!\cdots\!30}a+\frac{293041394894066}{509271609335533}$, $\frac{137839742752}{25\!\cdots\!65}a^{14}-\frac{3158667597972}{25\!\cdots\!65}a^{12}+\frac{14656462752744}{25\!\cdots\!65}a^{10}-\frac{50849024426861}{25\!\cdots\!65}a^{8}+\frac{603264148597566}{25\!\cdots\!65}a^{6}-\frac{153191546048622}{231487095152515}a^{4}+\frac{43\!\cdots\!93}{25\!\cdots\!65}a^{2}-\frac{267483881524280}{509271609335533}$, $\frac{17054095283267}{25\!\cdots\!50}a^{15}-\frac{5511507}{340172072230}a^{14}-\frac{375212438631999}{25\!\cdots\!50}a^{13}+\frac{146208987}{340172072230}a^{12}+\frac{757948858141504}{12\!\cdots\!25}a^{11}-\frac{1004403909}{340172072230}a^{10}-\frac{30\!\cdots\!11}{12\!\cdots\!25}a^{9}+\frac{3407009841}{340172072230}a^{8}+\frac{38\!\cdots\!39}{12\!\cdots\!25}a^{7}-\frac{15479240358}{170086036115}a^{6}-\frac{74\!\cdots\!02}{11\!\cdots\!75}a^{5}+\frac{81512887356}{170086036115}a^{4}+\frac{55\!\cdots\!77}{25\!\cdots\!50}a^{3}-\frac{109033291629}{170086036115}a^{2}-\frac{19\!\cdots\!53}{50\!\cdots\!30}a-\frac{4944892571}{68034414446}$, $\frac{21796349102809}{12\!\cdots\!25}a^{15}+\frac{1837169}{340172072230}a^{14}-\frac{479149949862768}{12\!\cdots\!25}a^{13}-\frac{48736329}{340172072230}a^{12}+\frac{38\!\cdots\!57}{25\!\cdots\!50}a^{11}+\frac{334801303}{340172072230}a^{10}-\frac{15\!\cdots\!83}{25\!\cdots\!50}a^{9}-\frac{1135669947}{340172072230}a^{8}+\frac{19\!\cdots\!37}{25\!\cdots\!50}a^{7}+\frac{5159746786}{170086036115}a^{6}-\frac{19\!\cdots\!53}{11\!\cdots\!75}a^{5}-\frac{27170962452}{170086036115}a^{4}+\frac{14\!\cdots\!33}{25\!\cdots\!50}a^{3}+\frac{36344430543}{170086036115}a^{2}-\frac{51\!\cdots\!07}{50\!\cdots\!30}a-\frac{61540731147}{34017207223}$, $\frac{2262418815981}{25\!\cdots\!65}a^{15}+\frac{616738359323}{50\!\cdots\!30}a^{14}-\frac{99138714029823}{50\!\cdots\!30}a^{13}-\frac{1363370113060}{509271609335533}a^{12}+\frac{196582489098444}{25\!\cdots\!65}a^{11}+\frac{5551620563178}{509271609335533}a^{10}-\frac{15\!\cdots\!09}{50\!\cdots\!30}a^{9}-\frac{20576403434524}{509271609335533}a^{8}+\frac{20\!\cdots\!53}{50\!\cdots\!30}a^{7}+\frac{13\!\cdots\!43}{25\!\cdots\!65}a^{6}-\frac{19\!\cdots\!19}{231487095152515}a^{5}-\frac{101243591249693}{92594838061006}a^{4}+\frac{72\!\cdots\!56}{25\!\cdots\!65}a^{3}+\frac{95\!\cdots\!58}{25\!\cdots\!65}a^{2}-\frac{24\!\cdots\!45}{509271609335533}a-\frac{339143300823823}{509271609335533}$, $\frac{26538602922351}{12\!\cdots\!25}a^{15}-\frac{49635045681}{509271609335533}a^{14}-\frac{583087461093537}{12\!\cdots\!25}a^{13}+\frac{1117540722897}{509271609335533}a^{12}+\frac{23\!\cdots\!49}{12\!\cdots\!25}a^{11}-\frac{4860123039655}{509271609335533}a^{10}-\frac{95\!\cdots\!61}{12\!\cdots\!25}a^{9}+\frac{16939186815437}{509271609335533}a^{8}+\frac{12\!\cdots\!59}{12\!\cdots\!25}a^{7}-\frac{210407031785744}{509271609335533}a^{6}-\frac{23\!\cdots\!02}{11\!\cdots\!75}a^{5}+\frac{46484746460580}{46297419030503}a^{4}+\frac{88\!\cdots\!56}{12\!\cdots\!25}a^{3}-\frac{15\!\cdots\!36}{509271609335533}a^{2}-\frac{31\!\cdots\!54}{25\!\cdots\!65}a+\frac{12\!\cdots\!51}{509271609335533}$, $\frac{73835434809861}{12\!\cdots\!25}a^{15}+\frac{122214853619246}{12\!\cdots\!25}a^{14}-\frac{14\!\cdots\!08}{12\!\cdots\!25}a^{13}-\frac{24\!\cdots\!16}{12\!\cdots\!25}a^{12}+\frac{700416974419907}{25\!\cdots\!65}a^{11}+\frac{11\!\cdots\!61}{25\!\cdots\!50}a^{10}-\frac{19\!\cdots\!74}{12\!\cdots\!25}a^{9}-\frac{32\!\cdots\!98}{12\!\cdots\!25}a^{8}+\frac{59\!\cdots\!49}{25\!\cdots\!50}a^{7}+\frac{98\!\cdots\!87}{25\!\cdots\!50}a^{6}-\frac{17\!\cdots\!91}{23\!\cdots\!50}a^{5}-\frac{17\!\cdots\!96}{11\!\cdots\!75}a^{4}+\frac{43\!\cdots\!79}{25\!\cdots\!50}a^{3}+\frac{29\!\cdots\!85}{10\!\cdots\!66}a^{2}+\frac{69\!\cdots\!49}{50\!\cdots\!30}a+\frac{145799014180577}{10\!\cdots\!66}$, $\frac{113601149548}{12\!\cdots\!25}a^{15}-\frac{478492070851}{25\!\cdots\!50}a^{14}-\frac{2618872099802}{12\!\cdots\!25}a^{13}+\frac{13151637481921}{25\!\cdots\!50}a^{12}+\frac{19700788882611}{25\!\cdots\!50}a^{11}-\frac{41672731584644}{12\!\cdots\!25}a^{10}-\frac{12968056088437}{25\!\cdots\!50}a^{9}+\frac{219612621789263}{25\!\cdots\!50}a^{8}+\frac{83381994682541}{25\!\cdots\!65}a^{7}-\frac{26\!\cdots\!01}{25\!\cdots\!50}a^{6}-\frac{206308324818989}{23\!\cdots\!50}a^{5}+\frac{500798442084888}{11\!\cdots\!75}a^{4}-\frac{25\!\cdots\!56}{12\!\cdots\!25}a^{3}-\frac{37\!\cdots\!31}{50\!\cdots\!30}a^{2}+\frac{293435708631589}{25\!\cdots\!65}a+\frac{1325076544963}{10\!\cdots\!66}$ (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: | \( 28231704.2902 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 28231704.2902 \cdot 1}{2\cdot\sqrt{43149301773875176172265625}}\cr\approx \mathstrut & 0.857392446714 \end{aligned}\] (assuming GRH)
Galois group
$C_4\wr C_2$ (as 16T42):
A solvable group of order 32 |
The 14 conjugacy class representatives for $C_4\wr C_2$ |
Character table for $C_4\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{505}) \), 4.4.2525.1 x2, 4.4.51005.1 x2, \(\Q(\sqrt{5}, \sqrt{101})\), 8.4.6568812813125.5, 8.4.643938125.1, 8.8.65037750625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 32 |
Degree 8 siblings: | 8.4.643938125.1, 8.4.6568812813125.5 |
Degree 16 sibling: | 16.0.17606641095812026885331265625.3 |
Minimal sibling: | 8.4.643938125.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(101\) | 101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
101.4.3.1 | $x^{4} + 404$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
101.4.3.1 | $x^{4} + 404$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |