Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 24*x^20 + 220*x^18 - 980*x^16 + 2267*x^14 - 2696*x^12 + 1324*x^10 + 265*x^8 - 563*x^6 + 215*x^4 - 29*x^2 + 1)
gp: K = bnfinit(y^22 - 24*y^20 + 220*y^18 - 980*y^16 + 2267*y^14 - 2696*y^12 + 1324*y^10 + 265*y^8 - 563*y^6 + 215*y^4 - 29*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 24*x^20 + 220*x^18 - 980*x^16 + 2267*x^14 - 2696*x^12 + 1324*x^10 + 265*x^8 - 563*x^6 + 215*x^4 - 29*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 24*x^20 + 220*x^18 - 980*x^16 + 2267*x^14 - 2696*x^12 + 1324*x^10 + 265*x^8 - 563*x^6 + 215*x^4 - 29*x^2 + 1)
\( x^{22} - 24 x^{20} + 220 x^{18} - 980 x^{16} + 2267 x^{14} - 2696 x^{12} + 1324 x^{10} + 265 x^{8} + \cdots + 1 \)
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: | $[20, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-7198079267989980836471065337135104\)
\(\medspace = -\,2^{22}\cdot 23^{20}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(34.59\) | 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: |
\(2\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{47}a^{18}+\frac{22}{47}a^{16}-\frac{2}{47}a^{14}-\frac{20}{47}a^{12}+\frac{8}{47}a^{10}-\frac{20}{47}a^{8}-\frac{21}{47}a^{6}+\frac{9}{47}a^{4}+\frac{9}{47}a^{2}+\frac{4}{47}$, $\frac{1}{47}a^{19}+\frac{22}{47}a^{17}-\frac{2}{47}a^{15}-\frac{20}{47}a^{13}+\frac{8}{47}a^{11}-\frac{20}{47}a^{9}-\frac{21}{47}a^{7}+\frac{9}{47}a^{5}+\frac{9}{47}a^{3}+\frac{4}{47}a$, $\frac{1}{302633}a^{20}+\frac{2913}{302633}a^{18}+\frac{81977}{302633}a^{16}-\frac{130399}{302633}a^{14}-\frac{148851}{302633}a^{12}+\frac{126602}{302633}a^{10}-\frac{104559}{302633}a^{8}+\frac{82977}{302633}a^{6}+\frac{83321}{302633}a^{4}-\frac{116105}{302633}a^{2}+\frac{66977}{302633}$, $\frac{1}{302633}a^{21}+\frac{2913}{302633}a^{19}+\frac{81977}{302633}a^{17}-\frac{130399}{302633}a^{15}-\frac{148851}{302633}a^{13}+\frac{126602}{302633}a^{11}-\frac{104559}{302633}a^{9}+\frac{82977}{302633}a^{7}+\frac{83321}{302633}a^{5}-\frac{116105}{302633}a^{3}+\frac{66977}{302633}a$
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: | $20$ | 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: |
$a^{21}-24a^{19}+220a^{17}-980a^{15}+2267a^{13}-2696a^{11}+1324a^{9}+265a^{7}-563a^{5}+215a^{3}-29a$, $\frac{236205}{302633}a^{20}-\frac{5537476}{302633}a^{18}+\frac{48836983}{302633}a^{16}-\frac{203250720}{302633}a^{14}+\frac{413606896}{302633}a^{12}-\frac{373543414}{302633}a^{10}+\frac{47995282}{302633}a^{8}+\frac{126288438}{302633}a^{6}-\frac{57202966}{302633}a^{4}+\frac{3639653}{302633}a^{2}-\frac{11643}{302633}$, $\frac{268802}{302633}a^{21}-\frac{5718640}{302633}a^{19}+\frac{42449447}{302633}a^{17}-\frac{122667134}{302633}a^{15}+\frac{63852819}{302633}a^{13}+\frac{266094116}{302633}a^{11}-\frac{395045478}{302633}a^{9}+\frac{97283477}{302633}a^{7}+\frac{97551324}{302633}a^{5}-\frac{53581639}{302633}a^{3}+\frac{6868504}{302633}a$, $\frac{340108}{302633}a^{21}-\frac{8076637}{302633}a^{19}+\frac{72724607}{302633}a^{17}-\frac{313673213}{302633}a^{15}+\frac{682100769}{302633}a^{13}-\frac{713344401}{302633}a^{11}+\frac{234104624}{302633}a^{9}+\frac{144513038}{302633}a^{7}-\frac{128319748}{302633}a^{5}+\frac{36647822}{302633}a^{3}-\frac{4801253}{302633}a$, $\frac{464664}{302633}a^{20}-\frac{10838551}{302633}a^{18}+\frac{95028558}{302633}a^{16}-\frac{393812569}{302633}a^{14}+\frac{808779246}{302633}a^{12}-\frac{786035731}{302633}a^{10}+\frac{219288252}{302633}a^{8}+\frac{183886762}{302633}a^{6}-\frac{147867155}{302633}a^{4}+\frac{32341216}{302633}a^{2}-\frac{1820501}{302633}$, $\frac{6765}{302633}a^{21}+\frac{589054}{302633}a^{19}-\frac{15508899}{302633}a^{17}+\frac{135103302}{302633}a^{15}-\frac{522641874}{302633}a^{13}+\frac{919300731}{302633}a^{11}-\frac{634587374}{302633}a^{9}-\frac{42239577}{302633}a^{7}+\frac{232425088}{302633}a^{5}-\frac{83502740}{302633}a^{3}+\frac{6812315}{302633}a$, $\frac{538310}{302633}a^{20}-\frac{12246057}{302633}a^{18}+\frac{103039587}{302633}a^{16}-\frac{397033980}{302633}a^{14}+\frac{709703205}{302633}a^{12}-\frac{506871945}{302633}a^{10}-\frac{32164663}{302633}a^{8}+\frac{198017998}{302633}a^{6}-\frac{66322830}{302633}a^{4}+\frac{553093}{302633}a^{2}+\frac{534804}{302633}$, $\frac{237364}{302633}a^{20}-\frac{5322858}{302633}a^{18}+\frac{43728315}{302633}a^{16}-\frac{161335502}{302633}a^{14}+\frac{264372011}{302633}a^{12}-\frac{152235198}{302633}a^{10}-\frac{37799855}{302633}a^{8}+\frac{57337065}{302633}a^{6}-\frac{12693242}{302633}a^{4}+\frac{2831305}{302633}a^{2}-\frac{702857}{302633}$, $\frac{213104}{302633}a^{20}-\frac{4751142}{302633}a^{18}+\frac{38693415}{302633}a^{16}-\frac{140944149}{302633}a^{14}+\frac{227835966}{302633}a^{12}-\frac{136672289}{302633}a^{10}-\frac{11867639}{302633}a^{8}+\frac{38215785}{302633}a^{6}-\frac{18634385}{302633}a^{4}+\frac{6455407}{302633}a^{2}-\frac{844202}{302633}$, $\frac{219416}{302633}a^{21}-\frac{4882850}{302633}a^{19}+\frac{39473318}{302633}a^{17}-\frac{139837076}{302633}a^{15}+\frac{199464905}{302633}a^{13}-\frac{28632579}{302633}a^{11}-\frac{169524779}{302633}a^{9}+\frac{102273521}{302633}a^{7}+\frac{21948142}{302633}a^{5}-\frac{25114985}{302633}a^{3}+\frac{3141379}{302633}a$, $\frac{448328}{302633}a^{21}+\frac{917993}{302633}a^{20}-\frac{10757402}{302633}a^{19}-\frac{21283986}{302633}a^{18}+\frac{98403103}{302633}a^{17}+\frac{184754175}{302633}a^{16}-\frac{435012719}{302633}a^{15}-\frac{752146033}{302633}a^{14}+\frac{982702612}{302633}a^{13}+\frac{1493197683}{302633}a^{12}-\frac{1088014059}{302633}a^{11}-\frac{1350118628}{302633}a^{10}+\frac{399003036}{302633}a^{9}+\frac{265207476}{302633}a^{8}+\frac{228438455}{302633}a^{7}+\frac{375532954}{302633}a^{6}-\frac{220510038}{302633}a^{5}-\frac{237058913}{302633}a^{4}+\frac{50164491}{302633}a^{3}+\frac{40261587}{302633}a^{2}-\frac{2833499}{302633}a-\frac{1213938}{302633}$, $\frac{340108}{302633}a^{21}-\frac{538310}{302633}a^{20}-\frac{8076637}{302633}a^{19}+\frac{12246057}{302633}a^{18}+\frac{72724607}{302633}a^{17}-\frac{103039587}{302633}a^{16}-\frac{313673213}{302633}a^{15}+\frac{397033980}{302633}a^{14}+\frac{682100769}{302633}a^{13}-\frac{709703205}{302633}a^{12}-\frac{713344401}{302633}a^{11}+\frac{506871945}{302633}a^{10}+\frac{234104624}{302633}a^{9}+\frac{32164663}{302633}a^{8}+\frac{144513038}{302633}a^{7}-\frac{198017998}{302633}a^{6}-\frac{128319748}{302633}a^{5}+\frac{66322830}{302633}a^{4}+\frac{36647822}{302633}a^{3}-\frac{553093}{302633}a^{2}-\frac{4801253}{302633}a-\frac{534804}{302633}$, $\frac{219416}{302633}a^{21}+\frac{151409}{302633}a^{20}-\frac{4882850}{302633}a^{19}-\frac{3223705}{302633}a^{18}+\frac{39473318}{302633}a^{17}+\frac{24095688}{302633}a^{16}-\frac{139837076}{302633}a^{15}-\frac{72411435}{302633}a^{14}+\frac{199464905}{302633}a^{13}+\frac{60785244}{302633}a^{12}-\frac{28632579}{302633}a^{11}+\frac{63963170}{302633}a^{10}-\frac{169524779}{302633}a^{9}-\frac{92184273}{302633}a^{8}+\frac{102273521}{302633}a^{7}-\frac{5823991}{302633}a^{6}+\frac{21948142}{302633}a^{5}+\frac{23782156}{302633}a^{4}-\frac{25114985}{302633}a^{3}-\frac{2230574}{302633}a^{2}+\frac{3444012}{302633}a-\frac{362749}{302633}$, $\frac{448328}{302633}a^{21}-\frac{237364}{302633}a^{20}-\frac{10757402}{302633}a^{19}+\frac{5322858}{302633}a^{18}+\frac{98403103}{302633}a^{17}-\frac{43728315}{302633}a^{16}-\frac{435012719}{302633}a^{15}+\frac{161335502}{302633}a^{14}+\frac{982702612}{302633}a^{13}-\frac{264372011}{302633}a^{12}-\frac{1088014059}{302633}a^{11}+\frac{152235198}{302633}a^{10}+\frac{399003036}{302633}a^{9}+\frac{37799855}{302633}a^{8}+\frac{228438455}{302633}a^{7}-\frac{57337065}{302633}a^{6}-\frac{220510038}{302633}a^{5}+\frac{12693242}{302633}a^{4}+\frac{50164491}{302633}a^{3}-\frac{2831305}{302633}a^{2}-\frac{3136132}{302633}a+\frac{702857}{302633}$, $\frac{268802}{302633}a^{21}-\frac{5718640}{302633}a^{19}+\frac{42449447}{302633}a^{17}-\frac{122667134}{302633}a^{15}+\frac{63852819}{302633}a^{13}+\frac{266094116}{302633}a^{11}-\frac{395045478}{302633}a^{9}+\frac{97283477}{302633}a^{7}+\frac{97551324}{302633}a^{5}-\frac{53581639}{302633}a^{3}+\frac{6868504}{302633}a+1$, $\frac{531545}{302633}a^{21}-\frac{213104}{302633}a^{20}-\frac{12835111}{302633}a^{19}+\frac{4751142}{302633}a^{18}+\frac{118548486}{302633}a^{17}-\frac{38693415}{302633}a^{16}-\frac{532137282}{302633}a^{15}+\frac{140944149}{302633}a^{14}+\frac{1232345079}{302633}a^{13}-\frac{227835966}{302633}a^{12}-\frac{1426172676}{302633}a^{11}+\frac{136672289}{302633}a^{10}+\frac{602422711}{302633}a^{9}+\frac{11867639}{302633}a^{8}+\frac{240257575}{302633}a^{7}-\frac{38215785}{302633}a^{6}-\frac{298747918}{302633}a^{5}+\frac{18634385}{302633}a^{4}+\frac{84055833}{302633}a^{3}-\frac{6455407}{302633}a^{2}-\frac{6277511}{302633}a+\frac{844202}{302633}$, $\frac{929781}{302633}a^{21}-\frac{21947946}{302633}a^{19}+\frac{195834767}{302633}a^{17}-\frac{832620231}{302633}a^{15}+\frac{1768662210}{302633}a^{13}-\frac{1769853253}{302633}a^{11}+\frac{469020153}{302633}a^{9}+\frac{470235030}{302633}a^{7}-\frac{333435660}{302633}a^{5}+\frac{54970607}{302633}a^{3}+\frac{364069}{302633}a+1$, $\frac{147382}{302633}a^{21}-\frac{433549}{302633}a^{20}-\frac{3834203}{302633}a^{19}+\frac{9836698}{302633}a^{18}+\frac{39091831}{302633}a^{17}-\frac{82432784}{302633}a^{16}-\frac{199314241}{302633}a^{15}+\frac{315653099}{302633}a^{14}+\frac{537371418}{302633}a^{13}-\frac{559361575}{302633}a^{12}-\frac{733377312}{302633}a^{11}+\frac{398068669}{302633}a^{10}+\frac{393367844}{302633}a^{9}+\frac{16465832}{302633}a^{8}+\frac{87757522}{302633}a^{7}-\frac{147510009}{302633}a^{6}-\frac{166880450}{302633}a^{5}+\frac{56264286}{302633}a^{4}+\frac{52442703}{302633}a^{3}-\frac{3449109}{302633}a^{2}-\frac{4373793}{302633}a-\frac{612875}{302633}$, $\frac{16789}{302633}a^{21}+\frac{432973}{302633}a^{20}-\frac{654626}{302633}a^{19}-\frac{10207469}{302633}a^{18}+\frac{9363665}{302633}a^{17}+\frac{90904943}{302633}a^{16}-\frac{63413644}{302633}a^{15}-\frac{385395019}{302633}a^{14}+\frac{214141991}{302633}a^{13}+\frac{815366228}{302633}a^{12}-\frac{344910835}{302633}a^{11}-\frac{812194031}{302633}a^{10}+\frac{217520061}{302633}a^{9}+\frac{215842427}{302633}a^{8}+\frac{24014917}{302633}a^{7}+\frac{209055916}{302633}a^{6}-\frac{79151108}{302633}a^{5}-\frac{148480234}{302633}a^{4}+\frac{28754638}{302633}a^{3}+\frac{27615702}{302633}a^{2}-\frac{3758288}{302633}a-\frac{1566210}{302633}$, $\frac{18012}{6439}a^{21}-\frac{300946}{302633}a^{20}-\frac{422771}{6439}a^{19}+\frac{6923199}{302633}a^{18}+\frac{3738072}{6439}a^{17}-\frac{59311272}{302633}a^{16}-\frac{15648844}{6439}a^{15}+\frac{235698478}{302633}a^{14}+\frac{32319569}{6439}a^{13}-\frac{445331194}{302633}a^{12}-\frac{30461588}{6439}a^{11}+\frac{354636747}{302633}a^{10}+\frac{5651074}{6439}a^{9}-\frac{5635192}{302633}a^{8}+\frac{9753530}{6439}a^{7}-\frac{140680933}{302633}a^{6}-\frac{5429740}{6439}a^{5}+\frac{53629588}{302633}a^{4}+\frac{448495}{6439}a^{3}+\frac{2278212}{302633}a^{2}+\frac{74584}{6439}a-\frac{1237661}{302633}$
| 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: | \( 3758789855.24 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{1}\cdot 3758789855.24 \cdot 1}{2\cdot\sqrt{7198079267989980836471065337135104}}\cr\approx \mathstrut & 0.145945100368 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 24*x^20 + 220*x^18 - 980*x^16 + 2267*x^14 - 2696*x^12 + 1324*x^10 + 265*x^8 - 563*x^6 + 215*x^4 - 29*x^2 + 1)
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 - 24*x^20 + 220*x^18 - 980*x^16 + 2267*x^14 - 2696*x^12 + 1324*x^10 + 265*x^8 - 563*x^6 + 215*x^4 - 29*x^2 + 1, 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 - 24*x^20 + 220*x^18 - 980*x^16 + 2267*x^14 - 2696*x^12 + 1324*x^10 + 265*x^8 - 563*x^6 + 215*x^4 - 29*x^2 + 1);
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 - 24*x^20 + 220*x^18 - 980*x^16 + 2267*x^14 - 2696*x^12 + 1324*x^10 + 265*x^8 - 563*x^6 + 215*x^4 - 29*x^2 + 1);
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
$C_{15}\times C_{420}$ (as 22T28):
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 22528 |
| The 208 conjugacy class representatives for $C_{15}\times C_{420}$ |
| Character table for $C_{15}\times C_{420}$ |
Intermediate fields
| \(\Q(\zeta_{23})^+\) |
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 siblings: | data not computed |
| Degree 44 siblings: | data not computed |
| Minimal sibling: | 22.12.7198079267989980836471065337135104.4 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $22$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | $22$ | $22$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | $22$ | R | ${\href{/padicField/29.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/37.11.0.1}{11} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | $22$ |
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 $22$ | $2$ | $11$ | $22$ | |||
|
\(23\)
| 23.22.20.1 | $x^{22} + 231 x^{21} + 24310 x^{20} + 1539615 x^{19} + 65271580 x^{18} + 1948237137 x^{17} + 41892053577 x^{16} + 652037968890 x^{15} + 7265319209115 x^{14} + 56290884204555 x^{13} + 287278948122936 x^{12} + 881069580352567 x^{11} + 1436394740619993 x^{10} + 1407272105659090 x^{9} + 908164935089445 x^{8} + 407525140102740 x^{7} + 130953632793951 x^{6} + 31291611122541 x^{5} + 17709125163290 x^{4} + 131485551467820 x^{3} + 905728303498040 x^{2} + 3760233666129578 x + 7096289029978197$ | $11$ | $2$ | $20$ | 22T1 | $[\ ]_{11}^{2}$ |