Properties

Label 444675bd1
Conductor 444675444675
Discriminant 2.742×1021-2.742\times 10^{21}
j-invariant 222985990144841995 -\frac{222985990144}{841995}
CM no
Rank 22
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+y=x3+x218725758x+31284762394y^2+y=x^3+x^2-18725758x+31284762394 Copy content Toggle raw display (homogenize, simplify)
y2z+yz2=x3+x2z18725758xz2+31284762394z3y^2z+yz^2=x^3+x^2z-18725758xz^2+31284762394z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x324268582800x+1459913097258000y^2=x^3-24268582800x+1459913097258000 Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, 1, 1, -18725758, 31284762394])
 
gp: E = ellinit([0, 1, 1, -18725758, 31284762394])
 
magma: E := EllipticCurve([0, 1, 1, -18725758, 31284762394]);
 
oscar: E = elliptic_curve([0, 1, 1, -18725758, 31284762394])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

ZZ\Z \oplus \Z

magma: MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(2207,26680)(2207, 26680)0.391047105201840508071553173430.39104710520184050807155317343\infty
(4748,222337)(4748, 222337)0.416724517566264711692444666410.41672451756626471169244466641\infty

Integral points

(4327,176962) \left(-4327, 176962\right) , (4327,176963) \left(-4327, -176963\right) , (3722,222337) \left(-3722, 222337\right) , (3722,222338) \left(-3722, -222338\right) , (1726,241798) \left(-1726, 241798\right) , (1726,241799) \left(-1726, -241799\right) , (1027,222337) \left(-1027, 222337\right) , (1027,222338) \left(-1027, -222338\right) , (397,196612) \left(-397, 196612\right) , (397,196613) \left(-397, -196613\right) , (1514,80041) \left(1514, 80041\right) , (1514,80042) \left(1514, -80042\right) , (2207,26680) \left(2207, 26680\right) , (2207,26681) \left(2207, -26681\right) , (2447,10999) \left(2447, 10999\right) , (2447,11000) \left(2447, -11000\right) , (2603,13612) \left(2603, 13612\right) , (2603,13613) \left(2603, -13613\right) , (2648,16537) \left(2648, 16537\right) , (2648,16538) \left(2648, -16538\right) , (3278,71662) \left(3278, 71662\right) , (3278,71663) \left(3278, -71663\right) , (4748,222337) \left(4748, 222337\right) , (4748,222338) \left(4748, -222338\right) , (13673,1526962) \left(13673, 1526962\right) , (13673,1526963) \left(13673, -1526963\right) , (19994,2765878) \left(19994, 2765878\right) , (19994,2765879) \left(19994, -2765879\right) , (31973,5667337) \left(31973, 5667337\right) , (31973,5667338) \left(31973, -5667338\right) , (443648,295486537) \left(443648, 295486537\right) , (443648,295486538) \left(443648, -295486538\right) Copy content Toggle raw display

comment: Integral points
 
sage: E.integral_points()
 
magma: IntegralPoints(E);
 

Invariants

Conductor: NN  =  444675 444675  = 352721123 \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: Δ\Delta  =  2742040655047461796875-2742040655047461796875 = 1375777117-1 \cdot 3^{7} \cdot 5^{7} \cdot 7^{7} \cdot 11^{7}
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  222985990144841995 -\frac{222985990144}{841995}  = 12123751711113793-1 \cdot 2^{12} \cdot 3^{-7} \cdot 5^{-1} \cdot 7^{-1} \cdot 11^{-1} \cdot 379^{3}
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: End(E)\mathrm{End}(E) = Z\Z
Geometric endomorphism ring: End(EQ)\mathrm{End}(E_{\overline{\Q}})  =  Z\Z    (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 2.97299031389757934734729622202.9729903138975793473472962220
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 0.0036313532463127645367316053160-0.0036313532463127645367316053160
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
abcabc quality: QQ ≈ 0.89082393515194340.8908239351519434
Szpiro ratio: σm\sigma_{m} ≈ 4.756314328949034.75631432894903

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 2 2
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: rr = 2 2
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) ≈ 0.157347506885167977851687780420.15734750688516797785168778042
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: Ω\Omega ≈ 0.144232082278261756980572478940.14423208227826175698057247894
comment: Real Period
 
sage: E.period_lattice().omega()
 
gp: if(E.disc>0,2,1)*E.omega[1]
 
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: pcp\prod_{p}c_p = 448 448  = 7222222 7\cdot2^{2}\cdot2^{2}\cdot2^{2}
comment: Tamagawa numbers
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
oscar: tamagawa_numbers(E)
 
Torsion order: #E(Q)tor\#E(\Q)_{\mathrm{tor}} = 11
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
Special value: L(2)(E,1)/2! L^{(2)}(E,1)/2! ≈ 10.16716223458472592727095537610.167162234584725927270955376
comment: Special L-value
 
r = E.rank();
 
E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Шan{}_{\mathrm{an}}  ≈  11    (rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

BSD formula

10.167162235L(2)(E,1)/2!=?#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.1442320.1573484481210.167162235\displaystyle 10.167162235 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.144232 \cdot 0.157348 \cdot 448}{1^2} \approx 10.167162235

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
 
E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;
 
Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();
 
omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();
 
assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
 
/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */
 
E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;
 
sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);
 
reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);
 
assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
 

Modular invariants

Modular form 444675.2.a.bd

q2q2+q3+2q42q6+q9+2q12+2q134q16q172q187q19+O(q20) q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} + q^{9} + 2 q^{12} + 2 q^{13} - 4 q^{16} - q^{17} - 2 q^{18} - 7 q^{19} + O(q^{20}) Copy content Toggle raw display

comment: q-expansion of modular form
 
sage: E.q_eigenform(20)
 
\\ actual modular form, use for small N
 
[mf,F] = mffromell(E)
 
Ser(mfcoefs(mf,20),q)
 
\\ or just the series
 
Ser(ellan(E,20),q)*q
 
magma: ModularForm(E);
 

For more coefficients, see the Downloads section to the right.

Modular degree: 46448640
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: yes
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 4 primes pp of bad reduction:

pp Tamagawa number Kodaira symbol Reduction type Root number ordp(N)\mathrm{ord}_p(N) ordp(Δ)\mathrm{ord}_p(\Delta) ordp(den(j))\mathrm{ord}_p(\mathrm{den}(j))
33 77 I7I_{7} split multiplicative -1 1 7 7
55 44 I1I_{1}^{*} additive 1 2 7 1
77 44 I1I_{1}^{*} additive -1 2 7 1
1111 44 I1I_{1}^{*} additive -1 2 7 1

comment: Local data
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
 

Galois representations

The \ell-adic Galois representation has maximal image for all primes \ell.

comment: mod p Galois image
 
sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

gens = [[1541, 2, 1541, 3], [211, 2, 211, 3], [661, 2, 661, 3], [1387, 2, 1387, 3], [1, 0, 2, 1], [1, 2, 0, 1], [2309, 2, 2308, 3], [1, 1, 2309, 0]]
 
GL(2,Integers(2310)).subgroup(gens)
 
Gens := [[1541, 2, 1541, 3], [211, 2, 211, 3], [661, 2, 661, 3], [1387, 2, 1387, 3], [1, 0, 2, 1], [1, 2, 0, 1], [2309, 2, 2308, 3], [1, 1, 2309, 0]];
 
sub<GL(2,Integers(2310))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 2310=235711 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 , index 22, genus 00, and generators

(1541215413),(21122113),(66126613),(1387213873),(1021),(1201),(2309223083),(1123090)\left(\begin{array}{rr} 1541 & 2 \\ 1541 & 3 \end{array}\right),\left(\begin{array}{rr} 211 & 2 \\ 211 & 3 \end{array}\right),\left(\begin{array}{rr} 661 & 2 \\ 661 & 3 \end{array}\right),\left(\begin{array}{rr} 1387 & 2 \\ 1387 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2309 & 2 \\ 2308 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 2309 & 0 \end{array}\right).

Input positive integer mm to see the generators of the reduction of HH to GL2(Z/mZ)\mathrm{GL}_2(\Z/m\Z):

The torsion field K:=Q(E[2310])K:=\Q(E[2310]) is a degree-18393661440001839366144000 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/2310Z)\GL_2(\Z/2310\Z).

The table below list all primes \ell for which the Serre invariants associated to the mod-\ell Galois representation are exceptional.

\ell Reduction type Serre weight Serre conductor
33 split multiplicative 44 148225=5272112 148225 = 5^{2} \cdot 7^{2} \cdot 11^{2}
55 additive 1818 17787=372112 17787 = 3 \cdot 7^{2} \cdot 11^{2}
77 additive 3232 3025=52112 3025 = 5^{2} \cdot 11^{2}
1111 additive 7272 3675=35272 3675 = 3 \cdot 5^{2} \cdot 7^{2}

Isogenies

gp: ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 444675bd consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 1155i1, its twist by 385385.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

pp-adic regulators

pp-adic regulators are not yet computed for curves that are not Γ0\Gamma_0-optimal.