LMFDB - Elliptic curve with Cremona label 442225bq2 (LMFDB label 442225.bq1)

Show commands: Magma / Oscar / PariGP / SageMath

Simplified equation

$y^2+y=x^3-16804550x+26540686156$ y^2+y=x^3-16804550x+26540686156	(homogenize, simplify)
$y^2z+yz^2=x^3-16804550xz^2+26540686156z^3$ y^2z+yz^2=x^3-16804550xz^2+26540686156z^3	(dehomogenize, simplify)
$y^2=x^3-268872800x+1698603914000$ y^2=x^3-268872800x+1698603914000	(homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 0, 1, -16804550, 26540686156])

gp: E = ellinit([0, 0, 1, -16804550, 26540686156])

magma: E := EllipticCurve([0, 0, 1, -16804550, 26540686156]);

oscar: E = elliptic_curve([0, 0, 1, -16804550, 26540686156])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

Invariants

Conductor:	$N$	=	$442225$	=	$5^{2} \cdot 7^{2} \cdot 19^{2}$	comment: Conductor sage: E.conductor().factor() gp: ellglobalred(E)[1] magma: Conductor(E); oscar: conductor(E)
Discriminant:	$\Delta$	=	$-593185702437524546875$	=	$-1 \cdot 5^{6} \cdot 7^{6} \cdot 19^{9}$	comment: Discriminant sage: E.discriminant().factor() gp: E.disc magma: Discriminant(E); oscar: discriminant(E)
j-invariant:	$j$	=	$-884736$	=	$-1 \cdot 2^{15} \cdot 3^{3}$	comment: j-invariant sage: E.j_invariant().factor() gp: E.j magma: jInvariant(E); oscar: j_invariant(E)
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z[(1+\sqrt{-19})/2]$ (potential complex multiplication)			sage: E.has_cm() magma: HasComplexMultiplication(E);
Sato-Tate group:	$\mathrm{ST}(E)$	=	$N(\mathrm{U}(1))$
Faltings height:	$h_{\mathrm{Faltings}}$	≈	$2.8997448604371522464532943162$			gp: ellheight(E) magma: FaltingsHeight(E); oscar: faltings_height(E)
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-1.0862584046823849384065322961$			magma: StableFaltingsHeight(E); oscar: stable_faltings_height(E)
$abc$ quality:	$Q$	≈	$1.3175706029138485$
Szpiro ratio:	$\sigma_{m}$	≈	$4.733003911598995$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$	sage: E.analytic_rank() gp: ellanalyticrank(E) magma: AnalyticRank(E);
Mordell-Weil rank:	$r$	=	$0$	comment: Rank sage: E.rank() gp: [lower,upper] = ellrank(E) magma: Rank(E);
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$	comment: Regulator sage: E.regulator() G = E.gen \\ if available matdet(ellheightmatrix(E,G)) magma: Regulator(E);
Real period:	$\Omega$	≈	$0.16250255969526091325918995013$	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:	$\prod_{p}c_p$	=	$2$ = $1\cdot1\cdot2$	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:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$	comment: Torsion order sage: E.torsion_order() gp: elltors(E)[1] magma: Order(TorsionSubgroup(E)); oscar: prod(torsion_structure(E)[1])
Special value:	$L(E,1)$	≈	$0.32500511939052182651837990026$	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:	Ш ${}_{\mathrm{an}}$	=	$1$ (exact)	comment: Order of Sha sage: E.sha().an_numerical() magma: MordellWeilShaInformation(E);

$\displaystyle 0.325005119 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.162503 \cdot 1.000000 \cdot 2}{1^2} \approx 0.325005119$

# 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 442225.2.a.bq

q - 2 q^{4} - 3 q^{9} - 5 q^{11} + 4 q^{16} - 7 q^{17} + O(q^{20})

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:	17556000	comment: Modular degree sage: E.modular_degree() gp: ellmoddegree(E) magma: ModularDegree(E);
$\Gamma_0(N)$ -optimal:	not computed^* (one of 2 curves in this isogeny class which might be optimal)
Manin constant:	1 (conditional^*)	comment: Manin constant magma: ManinConstant(E);

^* The optimal curve in each isogeny class has not been determined in all cases for conductors over 400000. The Manin constant is correct provided that curve 442225bq1 is optimal.

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$
$5$	$1$	$I_0^{*}$	additive	1	2	6
$7$	$1$	$I_0^{*}$	additive	-1	2	6
$19$	$2$	$III^{*}$	additive	1	2	9

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)];

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
$2$	good	$2$	$23275 = 5^{2} \cdot 7^{2} \cdot 19$
$5$	additive	$14$	$17689 = 7^{2} \cdot 19^{2}$
$7$	additive	$26$	$9025 = 5^{2} \cdot 19^{2}$
$19$	additive	$110$	$1225 = 5^{2} \cdot 7^{2}$

$p$ -adic regulators

All $p$ -adic regulators are identically $1$ since the rank is $0$ .

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

Mordell-Weil group structure

Invariants

BSD invariants

BSD formula

Modular invariants

Local data at primes of bad reduction

Galois representations

Isogenies

Twists

Iwasawa invariants

$p$ -adic regulators

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	442225bq2

Conductor	$442225$
Discriminant	$-5.932\times 10^{20}$
j-invariant	$-884736$
CM	yes ( $D=-19$ )
Rank	$0$
Torsion structure	trivial

Properties

Related objects

Downloads

Learn more

Mordell-Weil group structure

Invariants

Local data at primes of bad reduction

ppp-adic regulators

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

$p$ -adic regulators