Properties

Label 1225.b2
Conductor 12251225
Discriminant 6125-6125
j-invariant 9317 -9317
CM no
Rank 11
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+xy+y=x3+x28x+6y^2+xy+y=x^3+x^2-8x+6 Copy content Toggle raw display (homogenize, simplify)
y2z+xyz+yz2=x3+x2z8xz2+6z3y^2z+xyz+yz^2=x^3+x^2z-8xz^2+6z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x310395x+444150y^2=x^3-10395x+444150 Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, 1, 1, -8, 6])
 
Copy content gp:E = ellinit([1, 1, 1, -8, 6])
 
Copy content magma:E := EllipticCurve([1, 1, 1, -8, 6]);
 
Copy content oscar:E = elliptic_curve([1, 1, 1, -8, 6])
 
Copy content comment:Simplified equation
 
Copy content sage:E.short_weierstrass_model()
 
Copy content magma:WeierstrassModel(E);
 
Copy content oscar:short_weierstrass_model(E)
 

Mordell-Weil group structure

Z\Z

Copy content comment:Mordell-Weil group
 
Copy content magma:MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(0,2)(0, 2)0.175059577791148966853010517980.17505957779114896685301051798\infty

Integral points

(0,2) \left(0, 2\right) , (0,3) \left(0, -3\right) , (1,0) \left(1, 0\right) , (1,2) \left(1, -2\right) , (10,27) \left(10, 27\right) , (10,38) \left(10, -38\right) Copy content Toggle raw display

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

Invariants

Conductor: NN  =  1225 1225  = 52725^{2} \cdot 7^{2}
Copy content comment:Conductor
 
Copy content sage:E.conductor().factor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Copy content oscar:conductor(E)
 
Discriminant: Δ\Delta  =  6125-6125 = 15372-1 \cdot 5^{3} \cdot 7^{2}
Copy content comment:Discriminant
 
Copy content sage:E.discriminant().factor()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Copy content oscar:discriminant(E)
 
j-invariant: jj  =  9317 -9317  = 17113-1 \cdot 7 \cdot 11^{3}
Copy content comment:j-invariant
 
Copy content sage:E.j_invariant().factor()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Copy content 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)
Copy content comment:Potential complex multiplication
 
Copy content sage:E.has_cm()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 0.53497422781474375918341356667-0.53497422781474375918341356667
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 1.2616520640991544036844955239-1.2616520640991544036844955239
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
abcabc quality: QQ ≈ 0.80289886637937450.8028988663793745
Szpiro ratio: σm\sigma_{m} ≈ 2.53559803983214452.5355980398321445

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 1 1
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: rr = 1 1
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) ≈ 0.175059577791148966853010517980.17505957779114896685301051798
Copy content comment:Regulator
 
Copy content sage:E.regulator()
 
Copy content gp:G = E.gen \\ if available matdet(ellheightmatrix(E,G))
 
Copy content magma:Regulator(E);
 
Real period: Ω\Omega ≈ 4.13534856803743479387751597814.1353485680374347938775159781
Copy content comment:Real Period
 
Copy content sage:E.period_lattice().omega()
 
Copy content gp:if(E.disc>0,2,1)*E.omega[1]
 
Copy content magma:(Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: pcp\prod_{p}c_p = 2 2  = 21 2\cdot1
Copy content comment:Tamagawa numbers
 
Copy content sage:E.tamagawa_numbers()
 
Copy content gp:gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Copy content magma:TamagawaNumbers(E);
 
Copy content oscar:tamagawa_numbers(E)
 
Torsion order: #E(Q)tor\#E(\Q)_{\mathrm{tor}} = 11
Copy content comment:Torsion order
 
Copy content sage:E.torsion_order()
 
Copy content gp:elltors(E)[1]
 
Copy content magma:Order(TorsionSubgroup(E));
 
Copy content oscar:prod(torsion_structure(E)[1])
 
Special value: L(E,1) L'(E,1) ≈ 1.44786474867973160472234629491.4478647486797316047223462949
Copy content comment:Special L-value
 
Copy content sage:r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
Copy content gp:[r,L1r] = ellanalyticrank(E); L1r/r!
 
Copy content magma:Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Шan{}_{\mathrm{an}}  ≈  11    (rounded)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

1.447864749L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor214.1353490.1750602121.447864749\begin{aligned} 1.447864749 \approx L'(E,1) & = \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 4.135349 \cdot 0.175060 \cdot 2}{1^2} \\ & \approx 1.447864749\end{aligned}

Copy content comment:BSD formula
 
Copy content sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha) E = EllipticCurve([1, 1, 1, -8, 6]); 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)))
 
Copy content magma:/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */ E := EllipticCurve([1, 1, 1, -8, 6]); 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   1225.2.a.b

qq2q3q4+q6+3q82q9+q12+2q13q16+2q17+2q186q19+O(q20) q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} - 2 q^{9} + q^{12} + 2 q^{13} - q^{16} + 2 q^{17} + 2 q^{18} - 6 q^{19} + O(q^{20}) Copy content Toggle raw display

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:\\ 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
 
Copy content magma:ModularForm(E);
 

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

Modular degree: 48
Copy content comment:Modular degree
 
Copy content sage:E.modular_degree()
 
Copy content gp:ellmoddegree(E)
 
Copy content magma:ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: yes
Manin constant: 1
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 2 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))
55 22 IIIIII additive -1 2 3 0
77 11 IIII additive -1 2 2 0

Copy content comment:Local data
 
Copy content sage:E.local_data()
 
Copy content gp:ellglobalred(E)[5]
 
Copy content magma:[LocalInformation(E,p) : p in BadPrimes(E)];
 
Copy content 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 except those listed in the table below.

prime \ell mod-\ell image \ell-adic image
3737 37B.8.1 37.114.4.1

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

Copy content comment:Adelic image of Galois representation
 
Copy content sage:gens = [[101, 2072, 148, 2157], [1, 3256, 0, 1], [1, 0, 148, 1], [117, 2442, 37, 4323], [1, 0, 148, 3221], [1037, 0, 0, 2073], [1925, 3256, 1924, 1925], [4441, 0, 0, 3701], [1, 30, 0, 1], [75, 74, 1591, 2295], [1, 2590, 0, 1], [2591, 2590, 2590, 2591], [4848, 2923, 4477, 1259], [1, 0, 2590, 1]] GL(2,Integers(5180)).subgroup(gens)
 
Copy content magma:Gens := [[101, 2072, 148, 2157], [1, 3256, 0, 1], [1, 0, 148, 1], [117, 2442, 37, 4323], [1, 0, 148, 3221], [1037, 0, 0, 2073], [1925, 3256, 1924, 1925], [4441, 0, 0, 3701], [1, 30, 0, 1], [75, 74, 1591, 2295], [1, 2590, 0, 1], [2591, 2590, 2590, 2591], [4848, 2923, 4477, 1259], [1, 0, 2590, 1]]; sub<GL(2,Integers(5180))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 5180=225737 5180 = 2^{2} \cdot 5 \cdot 7 \cdot 37 , index 27362736, genus 9797, and generators

(10120721482157),(1325601),(101481),(1172442374323),(101483221),(1037002073),(1925325619241925),(4441003701),(13001),(757415912295),(1259001),(2591259025902591),(4848292344771259),(1025901)\left(\begin{array}{rr} 101 & 2072 \\ 148 & 2157 \end{array}\right),\left(\begin{array}{rr} 1 & 3256 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 148 & 1 \end{array}\right),\left(\begin{array}{rr} 117 & 2442 \\ 37 & 4323 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 148 & 3221 \end{array}\right),\left(\begin{array}{rr} 1037 & 0 \\ 0 & 2073 \end{array}\right),\left(\begin{array}{rr} 1925 & 3256 \\ 1924 & 1925 \end{array}\right),\left(\begin{array}{rr} 4441 & 0 \\ 0 & 3701 \end{array}\right),\left(\begin{array}{rr} 1 & 30 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 75 & 74 \\ 1591 & 2295 \end{array}\right),\left(\begin{array}{rr} 1 & 2590 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2591 & 2590 \\ 2590 & 2591 \end{array}\right),\left(\begin{array}{rr} 4848 & 2923 \\ 4477 & 1259 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2590 & 1 \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[5180])K:=\Q(E[5180]) is a degree-6186958848061869588480 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/5180Z)\GL_2(\Z/5180\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
22 good 22 245=572 245 = 5 \cdot 7^{2}
55 additive 1010 49=72 49 = 7^{2}
77 additive 1414 25=52 25 = 5^{2}

Isogenies

Copy content comment:Isogenies
 
Copy content gp:ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree dd for d=d= 37.
Its isogeny class 1225.b consists of 2 curves linked by isogenies of degree 37.

Twists

This elliptic curve is its own minimal quadratic twist.

Growth of torsion in number fields

The number fields KK of degree less than 24 such that E(K)torsE(K)_{\rm tors} is strictly larger than E(Q)torsE(\Q)_{\rm tors} (which is trivial) are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
33 3.1.980.1 Z/2Z\Z/2\Z not in database
66 6.0.19208000.2 Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
88 8.2.4020286921875.3 Z/3Z\Z/3\Z not in database
1212 deg 12 Z/4Z\Z/4\Z not in database
1212 Q(ζ35)+\Q(\zeta_{35})^+ Z/37Z\Z/37\Z not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

Iwasawa invariants

pp 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ord ord add add ss ord ord ord ord ord ord ord ord ord ss
λ\lambda-invariant(s) ? 1 - - 1,1 1 1 1 1 1 1 1 1 1 1,1
μ\mu-invariant(s) ? 0 - - 0,0 0 0 0 0 0 0 0 0 0 0,0

An entry ? indicates that the invariants have not yet been computed.

An entry - indicates that the invariants are not computed because the reduction is additive.

pp-adic regulators

Note: pp-adic regulator data only exists for primes p5p\ge 5 of good ordinary reduction.