LMFDB - Elliptic curve with Cremona label 432450cg2 (LMFDB label 432450.cg2)

Show commands: Magma / Oscar / PariGP / SageMath

Simplified equation

$y^2+xy=x^3-x^2+121261683x+192333546341$ y^2+xy=x^3-x^2+121261683x+192333546341	(homogenize, simplify)
$y^2z+xyz=x^3-x^2z+121261683xz^2+192333546341z^3$ y^2z+xyz=x^3-x^2z+121261683xz^2+192333546341z^3	(dehomogenize, simplify)
$y^2=x^3+1940186925x+12311287152750$ y^2=x^3+1940186925x+12311287152750	(homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, -1, 0, 121261683, 192333546341])

gp: E = ellinit([1, -1, 0, 121261683, 192333546341])

magma: E := EllipticCurve([1, -1, 0, 121261683, 192333546341]);

oscar: E = elliptic_curve([1, -1, 0, 121261683, 192333546341])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

Invariants

Conductor:	$N$	=	$432450$	=	$2 \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}$	comment: Conductor sage: E.conductor().factor() gp: ellglobalred(E)[1] magma: Conductor(E); oscar: conductor(E)
Discriminant:	$\Delta$	=	$-130102770747121823250000000$	=	$-1 \cdot 2^{7} \cdot 3^{9} \cdot 5^{9} \cdot 31^{9}$	comment: Discriminant sage: E.discriminant().factor() gp: E.disc magma: Discriminant(E); oscar: discriminant(E)
j-invariant:	$j$	=	$\frac{722458663317}{476656000}$	=	$2^{-7} \cdot 3^{6} \cdot 5^{-3} \cdot 31^{-3} \cdot 997^{3}$	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:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$ (no potential complex multiplication)			sage: E.has_cm() magma: HasComplexMultiplication(E);
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$
Faltings height:	$h_{\mathrm{Faltings}}$	≈	$3.6996683827561463831300384560$			gp: ellheight(E) magma: FaltingsHeight(E); oscar: faltings_height(E)
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$0.35399660779544080431864269942$			magma: StableFaltingsHeight(E); oscar: stable_faltings_height(E)
$abc$ quality:	$Q$	≈	$0.9749204621313927$
Szpiro ratio:	$\sigma_{m}$	≈	$5.197875738303996$

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:	$\Omega$	≈	$0.036679711268706012748310763436$	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:	$\prod_{p}c_p$	=	$16$ = $1\cdot2\cdot2\cdot2^{2}$	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:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$	comment: Torsion order sage: E.torsion_order() gp: elltors(E)[1] magma: Order(TorsionSubgroup(E)); oscar: prod(torsion_structure(E)[1])
Special value:	$L(E,1)$	≈	$0.58687538029929620397297221497$	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);

$\displaystyle 0.586875380 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.036680 \cdot 1.000000 \cdot 16}{1^2} \approx 0.586875380$

# 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 432450.2.a.cg

q - q^{2} + q^{4} + q^{7} - q^{8} - 3 q^{11} - 4 q^{13} - q^{14} + q^{16} - 6 q^{17} - q^{19} + 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:	139345920	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 432450cg1 is optimal.

Local data at primes of bad reduction

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

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$1$	$I_{7}$	nonsplit multiplicative	1	1	7	7
$3$	$2$	$III^{*}$	additive	1	2	9	0
$5$	$2$	$I_{3}^{*}$	additive	1	2	9	3
$31$	$4$	$I_{3}^{*}$	additive	-1	2	9	3

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 $\ell$ except those listed in the table below.

prime $\ell$	mod- $\ell$ image	$\ell$ -adic image
$3$	3B	3.4.0.1

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 = [[4, 3, 9, 7], [1861, 6, 1863, 19], [2634, 1079, 2477, 3546], [3715, 6, 3714, 7], [743, 3714, 2229, 3701], [2791, 6, 933, 19], [1, 6, 0, 1], [1919, 3714, 2037, 3701], [3, 4, 8, 11], [1, 0, 6, 1]]

GL(2,Integers(3720)).subgroup(gens)

Gens := [[4, 3, 9, 7], [1861, 6, 1863, 19], [2634, 1079, 2477, 3546], [3715, 6, 3714, 7], [743, 3714, 2229, 3701], [2791, 6, 933, 19], [1, 6, 0, 1], [1919, 3714, 2037, 3701], [3, 4, 8, 11], [1, 0, 6, 1]];

sub<GL(2,Integers(3720))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31$ , index $16$ , genus $0$ , and generators

$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1861 & 6 \\ 1863 & 19 \end{array}\right),\left(\begin{array}{rr} 2634 & 1079 \\ 2477 & 3546 \end{array}\right),\left(\begin{array}{rr} 3715 & 6 \\ 3714 & 7 \end{array}\right),\left(\begin{array}{rr} 743 & 3714 \\ 2229 & 3701 \end{array}\right),\left(\begin{array}{rr} 2791 & 6 \\ 933 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1919 & 3714 \\ 2037 & 3701 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$ .

The torsion field $K:=\Q(E[3720])$ is a degree- $1974730752000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3720\Z)$ .

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
$2$	nonsplit multiplicative	$4$	$72075 = 3 \cdot 5^{2} \cdot 31^{2}$
$3$	additive	$2$	$48050 = 2 \cdot 5^{2} \cdot 31^{2}$
$5$	additive	$18$	$17298 = 2 \cdot 3^{2} \cdot 31^{2}$
$7$	good	$2$	$216225 = 3^{2} \cdot 5^{2} \cdot 31^{2}$
$31$	additive	$512$	$450 = 2 \cdot 3^{2} \cdot 5^{2}$

$p$ -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	432450cg2

Conductor	$432450$
Discriminant	$-1.301\times 10^{26}$
j-invariant	$\frac{722458663317}{476656000}$
CM	no
Rank	$0$
Torsion structure	trivial

Properties

Related objects

Downloads

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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