Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 132*x^18 + 4400*x^14 - 3696*x^12 - 32384*x^10 + 9152*x^8 + 57728*x^6 + 30976*x^4 + 2816*x^2 - 256)
gp: K = bnfinit(y^22 - 132*y^18 + 4400*y^14 - 3696*y^12 - 32384*y^10 + 9152*y^8 + 57728*y^6 + 30976*y^4 + 2816*y^2 - 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 132*x^18 + 4400*x^14 - 3696*x^12 - 32384*x^10 + 9152*x^8 + 57728*x^6 + 30976*x^4 + 2816*x^2 - 256);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 132*x^18 + 4400*x^14 - 3696*x^12 - 32384*x^10 + 9152*x^8 + 57728*x^6 + 30976*x^4 + 2816*x^2 - 256)
\( x^{22} - 132 x^{18} + 4400 x^{14} - 3696 x^{12} - 32384 x^{10} + 9152 x^{8} + 57728 x^{6} + 30976 x^{4} + \cdots - 256 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[10, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(385779698695997532665150300037698289664\)
\(\medspace = 2^{24}\cdot 7^{10}\cdot 11^{22}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(56.74\) | 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: | not computed | ||
| 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: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}$, $\frac{1}{4}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{8}a^{9}$, $\frac{1}{8}a^{10}$, $\frac{1}{16}a^{11}$, $\frac{1}{32}a^{12}-\frac{1}{16}a^{10}-\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{13}-\frac{1}{8}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{32}a^{14}$, $\frac{1}{64}a^{15}-\frac{1}{16}a^{10}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{64}a^{16}-\frac{1}{4}a^{5}$, $\frac{1}{128}a^{17}-\frac{1}{16}a^{10}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{128}a^{18}-\frac{1}{8}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{128}a^{19}-\frac{1}{8}a^{8}$, $\frac{1}{8848289573632}a^{20}-\frac{5145653465}{4424144786816}a^{18}+\frac{10858496639}{2212072393408}a^{16}+\frac{10942544015}{1106036196704}a^{14}+\frac{2279459337}{1106036196704}a^{12}-\frac{3520000203}{553018098352}a^{10}-\frac{1}{16}a^{9}+\frac{31381050695}{276509049176}a^{8}-\frac{1}{8}a^{7}+\frac{41726000}{34563631147}a^{6}-\frac{7506759339}{138254524588}a^{4}-\frac{7320675873}{69127262294}a^{2}-\frac{1}{2}a+\frac{4523653492}{34563631147}$, $\frac{1}{8848289573632}a^{21}-\frac{5145653465}{4424144786816}a^{19}-\frac{12846637869}{4424144786816}a^{17}-\frac{12678543117}{2212072393408}a^{15}+\frac{2279459337}{1106036196704}a^{13}-\frac{3520000203}{553018098352}a^{11}-\frac{1}{16}a^{10}-\frac{795645113}{69127262294}a^{9}-\frac{1}{8}a^{8}+\frac{41726000}{34563631147}a^{7}-\frac{1}{8}a^{6}-\frac{7506759339}{138254524588}a^{5}-\frac{7320675873}{69127262294}a^{3}-\frac{1}{2}a^{2}-\frac{25516324163}{69127262294}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $15$ | 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: |
$\frac{11333079151}{1106036196704}a^{21}-\frac{16217373789}{2212072393408}a^{19}-\frac{745112141327}{553018098352}a^{17}+\frac{1066179508249}{1106036196704}a^{15}+\frac{24554906651851}{553018098352}a^{13}-\frac{38508409875199}{553018098352}a^{11}-\frac{78024771657495}{276509049176}a^{9}+\frac{40887944493937}{138254524588}a^{7}+\frac{26347179932459}{69127262294}a^{5}+\frac{1492782290427}{34563631147}a^{3}-\frac{180511716572}{34563631147}a$, $\frac{137942398433}{2212072393408}a^{21}-\frac{102967502089}{2212072393408}a^{19}-\frac{18137235760103}{2212072393408}a^{17}+\frac{1692753854177}{276509049176}a^{15}+\frac{149396593825351}{553018098352}a^{13}-\frac{239089426567871}{553018098352}a^{11}-\frac{236115460621775}{138254524588}a^{9}+\frac{64353535350495}{34563631147}a^{7}+\frac{78863468927654}{34563631147}a^{5}+\frac{11067941326791}{69127262294}a^{3}-\frac{1540236941216}{34563631147}a$, $\frac{38269559691}{2212072393408}a^{20}-\frac{3135168201}{276509049176}a^{18}-\frac{2517015620259}{1106036196704}a^{16}+\frac{1649420939111}{1106036196704}a^{14}+\frac{41519667528519}{553018098352}a^{12}-\frac{3909914950308}{34563631147}a^{10}-\frac{66913048314721}{138254524588}a^{8}+\frac{32676610413859}{69127262294}a^{6}+\frac{46694692597681}{69127262294}a^{4}+\frac{3623713179610}{34563631147}a^{2}-\frac{89646037973}{34563631147}$, $\frac{26684806701}{4424144786816}a^{21}-\frac{631305103}{553018098352}a^{19}-\frac{1761047684645}{2212072393408}a^{17}+\frac{166343908347}{1106036196704}a^{15}+\frac{1833877697351}{69127262294}a^{13}-\frac{15088706796857}{553018098352}a^{11}-\frac{26359704307373}{138254524588}a^{9}+\frac{12631129065657}{138254524588}a^{7}+\frac{23027128890861}{69127262294}a^{5}+\frac{4228989955206}{34563631147}a^{3}-\frac{255838630795}{34563631147}a$, $\frac{3423180781}{553018098352}a^{20}-\frac{9069180271}{2212072393408}a^{18}-\frac{900435335737}{1106036196704}a^{16}+\frac{298177134885}{553018098352}a^{14}+\frac{7423518911939}{276509049176}a^{12}-\frac{11240877512151}{276509049176}a^{10}-\frac{11934152185799}{69127262294}a^{8}+\frac{5859851038235}{34563631147}a^{6}+\frac{8374173532850}{34563631147}a^{4}+\frac{1230589551359}{34563631147}a^{2}-\frac{152350965135}{34563631147}$, $\frac{17981294793}{2212072393408}a^{20}-\frac{2970045933}{553018098352}a^{18}-\frac{1182271085993}{1106036196704}a^{16}+\frac{97801417551}{138254524588}a^{14}+\frac{19487068963605}{553018098352}a^{12}-\frac{1847881527453}{34563631147}a^{10}-\frac{31298168381141}{138254524588}a^{8}+\frac{31108445766555}{138254524588}a^{6}+\frac{21705036108195}{69127262294}a^{4}+\frac{1516030203433}{34563631147}a^{2}-\frac{104546058373}{34563631147}$, $\frac{64292412779}{4424144786816}a^{21}-\frac{24173008747}{2212072393408}a^{19}-\frac{264169804833}{138254524588}a^{17}+\frac{794710652205}{553018098352}a^{15}+\frac{17407294042895}{276509049176}a^{13}-\frac{55897129144883}{553018098352}a^{11}-\frac{109924510048573}{276509049176}a^{9}+\frac{60225193495461}{138254524588}a^{7}+\frac{18307947776299}{34563631147}a^{5}+\frac{1248591278678}{34563631147}a^{3}-\frac{361399422556}{34563631147}a$, $\frac{19974306603}{2212072393408}a^{21}+\frac{223581327787}{8848289573632}a^{20}-\frac{14898720617}{2212072393408}a^{19}-\frac{77256226961}{4424144786816}a^{18}-\frac{5252515298081}{4424144786816}a^{17}-\frac{7351028301849}{2212072393408}a^{16}+\frac{1958948424879}{2212072393408}a^{15}+\frac{2540551792529}{1106036196704}a^{14}+\frac{21631131338533}{553018098352}a^{13}+\frac{121185520984369}{1106036196704}a^{12}-\frac{34594106957137}{553018098352}a^{11}-\frac{93547849552085}{553018098352}a^{10}-\frac{136691216043673}{553018098352}a^{9}-\frac{193744536770853}{276509049176}a^{8}+\frac{74311965866783}{276509049176}a^{7}+\frac{197864664899259}{276509049176}a^{6}+\frac{11390258564047}{34563631147}a^{5}+\frac{132813677839505}{138254524588}a^{4}+\frac{1801964279825}{69127262294}a^{3}+\frac{8220713068917}{69127262294}a^{2}-\frac{198321467965}{34563631147}a-\frac{300238697756}{34563631147}$, $\frac{19974306603}{2212072393408}a^{21}-\frac{223581327787}{8848289573632}a^{20}-\frac{14898720617}{2212072393408}a^{19}+\frac{77256226961}{4424144786816}a^{18}-\frac{5252515298081}{4424144786816}a^{17}+\frac{7351028301849}{2212072393408}a^{16}+\frac{1958948424879}{2212072393408}a^{15}-\frac{2540551792529}{1106036196704}a^{14}+\frac{21631131338533}{553018098352}a^{13}-\frac{121185520984369}{1106036196704}a^{12}-\frac{34594106957137}{553018098352}a^{11}+\frac{93547849552085}{553018098352}a^{10}-\frac{136691216043673}{553018098352}a^{9}+\frac{193744536770853}{276509049176}a^{8}+\frac{74311965866783}{276509049176}a^{7}-\frac{197864664899259}{276509049176}a^{6}+\frac{11390258564047}{34563631147}a^{5}-\frac{132813677839505}{138254524588}a^{4}+\frac{1801964279825}{69127262294}a^{3}-\frac{8220713068917}{69127262294}a^{2}-\frac{198321467965}{34563631147}a+\frac{300238697756}{34563631147}$, $\frac{51503277247}{4424144786816}a^{21}-\frac{11033132583}{1106036196704}a^{19}-\frac{3383759966695}{2212072393408}a^{17}+\frac{725811479301}{553018098352}a^{15}+\frac{27819462371371}{553018098352}a^{13}-\frac{47750464407803}{553018098352}a^{11}-\frac{10715055274953}{34563631147}a^{9}+\frac{53123052098845}{138254524588}a^{7}+\frac{26889030700435}{69127262294}a^{5}-\frac{759072423326}{34563631147}a^{3}-\frac{493030139223}{34563631147}a$, $\frac{20521334561}{4424144786816}a^{20}-\frac{8465340127}{2212072393408}a^{18}-\frac{674393465207}{1106036196704}a^{16}+\frac{555421589735}{1106036196704}a^{14}+\frac{2775085974867}{138254524588}a^{12}-\frac{9293865559071}{276509049176}a^{10}-\frac{4309975536349}{34563631147}a^{8}+\frac{10106129859323}{69127262294}a^{6}+\frac{5489573472829}{34563631147}a^{4}+\frac{213061863875}{34563631147}a^{2}-\frac{130893960799}{34563631147}$, $\frac{21096666301}{8848289573632}a^{21}-\frac{274831876967}{8848289573632}a^{20}-\frac{16847303309}{4424144786816}a^{19}+\frac{102822450305}{4424144786816}a^{18}-\frac{1381596481195}{4424144786816}a^{17}+\frac{9032293765933}{2212072393408}a^{16}+\frac{276667600197}{553018098352}a^{15}-\frac{3381162617913}{1106036196704}a^{14}+\frac{5623155982157}{553018098352}a^{13}-\frac{74359531814527}{553018098352}a^{12}-\frac{6985874533493}{276509049176}a^{11}+\frac{7452628886631}{34563631147}a^{10}-\frac{28940522729897}{553018098352}a^{9}+\frac{58594654995693}{69127262294}a^{8}+\frac{8771050042235}{69127262294}a^{7}-\frac{256052376189137}{276509049176}a^{6}+\frac{4565747350581}{138254524588}a^{5}-\frac{77601358299291}{69127262294}a^{4}-\frac{2417477555139}{34563631147}a^{3}-\frac{6493305570497}{69127262294}a^{2}-\frac{1030625224303}{69127262294}a+\frac{624941609040}{34563631147}$, $\frac{57307710365}{4424144786816}a^{21}+\frac{204575799657}{8848289573632}a^{20}-\frac{37889066429}{4424144786816}a^{19}-\frac{71404040859}{4424144786816}a^{18}-\frac{942249396821}{553018098352}a^{17}-\frac{6726021295989}{2212072393408}a^{16}+\frac{622858117005}{553018098352}a^{15}+\frac{2347759486911}{1106036196704}a^{14}+\frac{62163454567213}{1106036196704}a^{13}+\frac{110874731900935}{1106036196704}a^{12}-\frac{47010734099499}{553018098352}a^{11}-\frac{85963530506003}{553018098352}a^{10}-\frac{200060685776679}{553018098352}a^{9}-\frac{177015529307901}{276509049176}a^{8}+\frac{98276479940315}{276509049176}a^{7}+\frac{22761375112170}{34563631147}a^{6}+\frac{17381231296654}{34563631147}a^{5}+\frac{120962375949891}{138254524588}a^{4}+\frac{2742905608829}{34563631147}a^{3}+\frac{3695521376989}{34563631147}a^{2}+\frac{73906281783}{69127262294}a-\frac{204715747541}{34563631147}$, $\frac{70689056473}{8848289573632}a^{21}+\frac{1450465079}{2212072393408}a^{20}-\frac{20153981693}{4424144786816}a^{19}-\frac{1454104367}{4424144786816}a^{18}-\frac{4650523969949}{4424144786816}a^{17}-\frac{191059621509}{2212072393408}a^{16}+\frac{1325819613857}{2212072393408}a^{15}+\frac{46961554731}{1106036196704}a^{14}+\frac{19194486559137}{553018098352}a^{13}+\frac{3162429549319}{1106036196704}a^{12}-\frac{27268888182051}{553018098352}a^{11}-\frac{1039766310989}{276509049176}a^{10}-\frac{15718852400557}{69127262294}a^{9}-\frac{5254740012429}{276509049176}a^{8}+\frac{27530681051261}{138254524588}a^{7}+\frac{3686248895029}{276509049176}a^{6}+\frac{22669132610493}{69127262294}a^{5}+\frac{976842537620}{34563631147}a^{4}+\frac{5208159854167}{69127262294}a^{3}+\frac{365292452636}{34563631147}a^{2}+\frac{133257965689}{34563631147}a+\frac{40197209953}{34563631147}$, $\frac{104181712323}{4424144786816}a^{21}+\frac{59252943103}{4424144786816}a^{20}-\frac{31869624957}{2212072393408}a^{19}-\frac{40984854607}{4424144786816}a^{18}-\frac{13706743208387}{4424144786816}a^{17}-\frac{3894028117791}{2212072393408}a^{16}+\frac{4196070116407}{2212072393408}a^{15}+\frac{168941418963}{138254524588}a^{14}+\frac{113120893516697}{1106036196704}a^{13}+\frac{64098710768553}{1106036196704}a^{12}-\frac{41409299216391}{276509049176}a^{11}-\frac{49818789999505}{553018098352}a^{10}-\frac{184123254571815}{276509049176}a^{9}-\frac{12737205079921}{34563631147}a^{8}+\frac{85667841374779}{138254524588}a^{7}+\frac{106149588333729}{276509049176}a^{6}+\frac{131939687427243}{138254524588}a^{5}+\frac{34512017253239}{69127262294}a^{4}+\frac{11568065881821}{69127262294}a^{3}+\frac{3783422099983}{69127262294}a^{2}-\frac{441372710066}{34563631147}a-\frac{143325874329}{34563631147}$
| 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: | \( 549685991097 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 549685991097 \cdot 1}{2\cdot\sqrt{385779698695997532665150300037698289664}}\cr\approx \mathstrut & 0.881645825133212 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 132*x^18 + 4400*x^14 - 3696*x^12 - 32384*x^10 + 9152*x^8 + 57728*x^6 + 30976*x^4 + 2816*x^2 - 256)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^22 - 132*x^18 + 4400*x^14 - 3696*x^12 - 32384*x^10 + 9152*x^8 + 57728*x^6 + 30976*x^4 + 2816*x^2 - 256, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^22 - 132*x^18 + 4400*x^14 - 3696*x^12 - 32384*x^10 + 9152*x^8 + 57728*x^6 + 30976*x^4 + 2816*x^2 - 256);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 132*x^18 + 4400*x^14 - 3696*x^12 - 32384*x^10 + 9152*x^8 + 57728*x^6 + 30976*x^4 + 2816*x^2 - 256);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_2^{10}.F_{11}$ (as 22T34):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 112640 |
| The 44 conjugacy class representatives for $C_2^{10}.F_{11}$ |
| Character table for $C_2^{10}.F_{11}$ |
Intermediate fields
| 11.11.4910318845910094848.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 22 sibling: | data not computed |
| Degree 44 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 | $20{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | $20{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.5.0.1}{5} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $22$ | $1$ | $24$ | |||
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.20.10.1 | $x^{20} + 70 x^{18} + 2207 x^{16} + 2 x^{15} + 41168 x^{14} - 68 x^{13} + 501639 x^{12} - 3674 x^{11} + 4175501 x^{10} - 48430 x^{9} + 24202032 x^{8} - 163712 x^{7} + 97377995 x^{6} + 430996 x^{5} + 259701777 x^{4} + 2947158 x^{3} + 412861211 x^{2} + 7541370 x + 287825400$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ | |
|
\(11\)
| 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |
| 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |