Normalized defining polynomial
\( x^{20} - 120 x^{18} + 7888 x^{16} - 1764 x^{15} - 45000 x^{14} - 6315120 x^{13} + 54010144 x^{12} + 90853056 x^{11} - 3602584704 x^{10} + 14654077200 x^{9} + 112702188032 x^{8} - 1752566231808 x^{7} + 11914781620224 x^{6} - 54255565782264 x^{5} + 184031339300816 x^{4} - 477951506932032 x^{3} + 893163408031104 x^{2} - 1066289337176448 x + 709016550047036 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(5395000787036562175590777878325808267813453824000000000000000=2^{38}\cdot 3^{16}\cdot 5^{15}\cdot 7^{17}\cdot 11^{10}\cdot 19^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $1087.93$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 5, 7, 11, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7} a^{9} - \frac{2}{7} a^{7} - \frac{1}{7} a^{5} + \frac{1}{7} a^{3}$, $\frac{1}{28} a^{10} + \frac{3}{7} a^{8} - \frac{2}{7} a^{6} + \frac{2}{7} a^{4} - \frac{1}{2}$, $\frac{1}{28} a^{11} - \frac{3}{7} a^{7} - \frac{2}{7} a^{5} - \frac{3}{7} a^{3} - \frac{1}{2} a$, $\frac{1}{28} a^{12} - \frac{3}{7} a^{8} - \frac{2}{7} a^{6} - \frac{3}{7} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{28} a^{13} - \frac{1}{7} a^{7} + \frac{1}{7} a^{5} - \frac{1}{14} a^{3}$, $\frac{1}{28} a^{14} - \frac{1}{7} a^{8} + \frac{1}{7} a^{6} - \frac{1}{14} a^{4}$, $\frac{1}{2156} a^{15} + \frac{1}{154} a^{14} - \frac{1}{98} a^{13} - \frac{1}{77} a^{12} - \frac{9}{539} a^{11} - \frac{3}{308} a^{10} - \frac{36}{539} a^{9} - \frac{20}{77} a^{8} - \frac{128}{539} a^{7} - \frac{26}{77} a^{6} - \frac{13}{1078} a^{5} - \frac{9}{77} a^{4} + \frac{17}{77} a^{3} + \frac{4}{11} a^{2} - \frac{1}{2}$, $\frac{1}{58212} a^{16} + \frac{2}{14553} a^{15} + \frac{433}{58212} a^{14} + \frac{437}{29106} a^{13} - \frac{31}{1764} a^{12} - \frac{89}{19404} a^{11} - \frac{197}{14553} a^{10} + \frac{1000}{14553} a^{9} + \frac{2419}{4851} a^{8} - \frac{5497}{14553} a^{7} - \frac{3065}{29106} a^{6} + \frac{1147}{4851} a^{5} - \frac{103}{1386} a^{4} + \frac{971}{2079} a^{3} + \frac{73}{594} a^{2} + \frac{1}{54} a + \frac{10}{27}$, $\frac{1}{1571724} a^{17} + \frac{1}{261954} a^{16} - \frac{8}{130977} a^{15} + \frac{7379}{1571724} a^{14} - \frac{6707}{392931} a^{13} - \frac{2293}{130977} a^{12} + \frac{1895}{112266} a^{11} + \frac{2977}{785862} a^{10} + \frac{9172}{392931} a^{9} - \frac{66694}{392931} a^{8} - \frac{90913}{785862} a^{7} - \frac{97633}{392931} a^{6} + \frac{5387}{130977} a^{5} - \frac{41963}{112266} a^{4} - \frac{13526}{56133} a^{3} - \frac{13}{33} a^{2} - \frac{2}{81} a - \frac{74}{729}$, $\frac{1}{16338070980} a^{18} + \frac{13}{47632860} a^{17} + \frac{989}{151278435} a^{16} + \frac{494441}{2334010140} a^{15} + \frac{13873561}{5446023660} a^{14} + \frac{6413783}{466802028} a^{13} + \frac{27339787}{4084517745} a^{12} - \frac{1364501}{259334460} a^{11} - \frac{4642885}{272301183} a^{10} - \frac{456338}{194500845} a^{9} - \frac{781313927}{2723011830} a^{8} - \frac{17077331}{77800338} a^{7} + \frac{1959571}{116700507} a^{6} + \frac{76966333}{1167005070} a^{5} - \frac{24919877}{55571670} a^{4} + \frac{508357}{15155910} a^{3} + \frac{125219}{1323135} a^{2} - \frac{592063}{2165130} a + \frac{33532}{1082565}$, $\frac{1}{181897886133986808411550218526579252881929417000539878323346536748527503315918977270931052817091319856300} a^{19} - \frac{4311323171238195074361008118881857970902855302374445120942039839552903572414165387186195696693}{181897886133986808411550218526579252881929417000539878323346536748527503315918977270931052817091319856300} a^{18} - \frac{4637894673279945397237919271256423569807419787691183079947657318272475773544969587768016890508574}{45474471533496702102887554631644813220482354250134969580836634187131875828979744317732763204272829964075} a^{17} + \frac{242425875415220650876179572986367639110209226489817994222096437311102599081107865776649659760239153}{181897886133986808411550218526579252881929417000539878323346536748527503315918977270931052817091319856300} a^{16} - \frac{1701058207749882413884717830888204014121164233523714576491473520060331215967109749597155114407651679}{45474471533496702102887554631644813220482354250134969580836634187131875828979744317732763204272829964075} a^{15} - \frac{640507074652408924240722492096501246401511183493560831953124695422869121587059998849646073067962641563}{90948943066993404205775109263289626440964708500269939161673268374263751657959488635465526408545659928150} a^{14} + \frac{9951563896358225582887688179484090871760600879948118268895592488743970381122568252863095652780312569}{8268085733363036745979555387571784221905882590933630832879388034023977423450862603224138764413241811650} a^{13} + \frac{2437038999388500620387490768427058404786989567639103000613095535826170395364099697142161701766300478681}{181897886133986808411550218526579252881929417000539878323346536748527503315918977270931052817091319856300} a^{12} - \frac{724656986636905575144868361233060399285226065446374637894822156290915352140311775369760525521232881513}{60632628711328936137183406175526417627309805666846626107782178916175834438639659090310350939030439952100} a^{11} + \frac{60177894265540966391633022996938339770907840513386584568340552060336658560456301214861444557105206743}{30316314355664468068591703087763208813654902833423313053891089458087917219319829545155175469515219976050} a^{10} - \frac{246936921732950398071396143641629671949410135950873370155223947721530771374556412080023259859404002061}{10105438118554822689530567695921069604551634277807771017963696486029305739773276515051725156505073325350} a^{9} + \frac{347766923970818158818140455854685574263294774321363266913168890214498017529755702531109600609150980823}{10105438118554822689530567695921069604551634277807771017963696486029305739773276515051725156505073325350} a^{8} + \frac{443250538011024329670076135165331371592237025478012680129843082659782767565807802666277132834205916907}{1299270615242762917225358703761280377728067264289570559452475262489482166542278409078078948693509427545} a^{7} + \frac{568763538408963332890314323918391535744371634487106888123680600421501693687469680580661806358064813463}{1181155104766148106568507912510254888843697512990518690411341147717711060492980371889162680630463115950} a^{6} - \frac{2994034884410182549594872861760775853368696068366856643678277313421239376332371299258836079513626602879}{6496353076213814586126793518806401888640336321447852797262376312447410832711392045390394743467547137725} a^{5} - \frac{221344910005285756887550952576713231377232894457345357538983118460126686116094206776130112539721227738}{928050439459116369446684788400914555520048045921121828180339473206772976101627435055770677638221019675} a^{4} + \frac{283069594790450162333639293917046277257509789894832142038232143612934835476222607279537293518166516}{185610087891823273889336957680182911104009609184224365636067894641354595220325487011154135527644203935} a^{3} + \frac{27876063418264988843849339938323592572061399213816888157260860284569552474881973848648523802971690573}{265157268416890391270481368114547015862870870263177665194382706630506564600464981444505907896634577050} a^{2} + \frac{7515047569880472245777392830415730145940412990371522844542225191389857729133256390097927893356107}{964208248788692331892659520416534603137712255502464237070482569565478416728963568889112392351398462} a + \frac{2319923778059292118366196200262466305059483223934710890020972591929085582374037986836784109567989731}{12052603109858654148658244005206682539221403193780802963381032119568480209112044611113904904392480775}$
Class group and class number
Not computed
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$D_4\times F_5$ (as 20T42):
| A solvable group of order 160 |
| The 25 conjugacy class representatives for $D_4\times F_5$ |
| Character table for $D_4\times F_5$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-110}) \), 4.0.128744000.6, 5.1.388962000.4, 10.0.249504125867966914560000000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
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 | R | R | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.19.1 | $x^{10} - 2$ | $10$ | $1$ | $19$ | $F_{5}\times C_2$ | $[3]_{5}^{4}$ |
| 2.10.19.1 | $x^{10} - 2$ | $10$ | $1$ | $19$ | $F_{5}\times C_2$ | $[3]_{5}^{4}$ | |
| $3$ | 3.5.4.1 | $x^{5} - 3$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |
| 3.5.4.1 | $x^{5} - 3$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 3.10.8.1 | $x^{10} - 3 x^{5} + 18$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 5 | Data not computed | ||||||
| $7$ | 7.10.9.2 | $x^{10} + 14$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $[\ ]_{10}^{4}$ |
| 7.10.8.1 | $x^{10} - 7 x^{5} + 147$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| $11$ | 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $19$ | $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |