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Properties

Label 438702.l2
Conductor $438702$
Discriminant $-1.177\times 10^{20}$
j-invariant \( \frac{8361347972165063}{4875084310338} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3+x^2+1222031x+45570607\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3+x^2z+1222031xz^2+45570607z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+1583751501x+2102385964302\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 1, 0, 1222031, 45570607])
 
gp: E = ellinit([1, 1, 0, 1222031, 45570607])
 
magma: E := EllipticCurve([1, 1, 0, 1222031, 45570607]);
 
oscar: E = elliptic_curve([1, 1, 0, 1222031, 45570607])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z \oplus \Z \oplus \Z/{2}\Z\)

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generators and heights

$P$ =  \(\left(961, 45439\right)\) Copy content Toggle raw display \(\left(\frac{37999}{4}, \frac{7419935}{8}\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $1.7530220292116101604133098823$$2.2359425128342324436333371074$

sage: E.gens()
 
magma: Generators(E);
 
gp: E.gen
 

Torsion generators

\( \left(-\frac{149}{4}, \frac{149}{8}\right) \) Copy content Toggle raw display

comment: Torsion subgroup
 
sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 
oscar: torsion_structure(E)
 

Integral points

\( \left(-31, 2783\right) \), \( \left(-31, -2752\right) \), \( \left(961, 45439\right) \), \( \left(961, -46400\right) \), \( \left(1191, 55904\right) \), \( \left(1191, -57095\right) \), \( \left(3469, 212848\right) \), \( \left(3469, -216317\right) \) Copy content Toggle raw display

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

Invariants

Conductor: \( 438702 \)  =  $2 \cdot 3 \cdot 11 \cdot 17^{2} \cdot 23$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-117672683921600888322 $  =  $-1 \cdot 2 \cdot 3^{2} \cdot 11^{6} \cdot 17^{8} \cdot 23^{2} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{8361347972165063}{4875084310338} \)  =  $2^{-1} \cdot 3^{-2} \cdot 11^{-6} \cdot 17^{-2} \cdot 23^{-2} \cdot 202967^{3}$
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.5406315217639553836653922383\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $1.1240248497358473435406249294\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $0.9583230676909829\dots$
Szpiro ratio: $4.130497914644972\dots$

BSD invariants

Analytic rank: $2$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $3.6400126898445312690400016729\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.11274232922283994928968293121\dots$
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: $ 96 $  = $ 1\cdot2\cdot( 2 \cdot 3 )\cdot2^{2}\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: $2$
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
Analytic order of Ш: $1$ ( rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Special value: $ L^{(2)}(E,1)/2! $ ≈ $ 9.8492042172904163136876742636 $
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);
 

BSD formula

$\displaystyle 9.849204217 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.112742 \cdot 3.640013 \cdot 96}{2^2} \approx 9.849204217$

# 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 438702.2.a.l

\( q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} - 4 q^{7} - q^{8} + q^{9} - 2 q^{10} + q^{11} - q^{12} - 4 q^{13} + 4 q^{14} - 2 q^{15} + q^{16} - q^{18} + O(q^{20}) \) 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: 19243008
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: no
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data

This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $v_p(N)$ $v_p(\Delta)$ $v_p(\mathrm{den}(j))$
$2$ $1$ $I_{1}$ nonsplit multiplicative 1 1 1 1
$3$ $2$ $I_{2}$ nonsplit multiplicative 1 1 2 2
$11$ $6$ $I_{6}$ split multiplicative -1 1 6 6
$17$ $4$ $I_{2}^{*}$ additive 1 2 8 2
$23$ $2$ $I_{2}$ nonsplit multiplicative 1 1 2 2

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
$2$ 2B 8.6.0.5

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 = [[409, 4, 818, 9], [1057, 4, 2114, 9], [1, 2, 2, 5], [4485, 4, 4484, 5], [1, 4, 0, 1], [2993, 4, 1498, 9], [1, 0, 4, 1], [1684, 2809, 561, 3928], [2, 1, 2243, 0], [3, 4, 8, 11]]
 
GL(2,Integers(4488)).subgroup(gens)
 
Gens := [[409, 4, 818, 9], [1057, 4, 2114, 9], [1, 2, 2, 5], [4485, 4, 4484, 5], [1, 4, 0, 1], [2993, 4, 1498, 9], [1, 0, 4, 1], [1684, 2809, 561, 3928], [2, 1, 2243, 0], [3, 4, 8, 11]];
 
sub<GL(2,Integers(4488))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4488 = 2^{3} \cdot 3 \cdot 11 \cdot 17 \), index $12$, genus $0$, and generators

$\left(\begin{array}{rr} 409 & 4 \\ 818 & 9 \end{array}\right),\left(\begin{array}{rr} 1057 & 4 \\ 2114 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 4485 & 4 \\ 4484 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2993 & 4 \\ 1498 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1684 & 2809 \\ 561 & 3928 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 2243 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[4488])$ is a degree-$6353112268800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4488\Z)$.

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 $4$ \( 289 = 17^{2} \)
$3$ nonsplit multiplicative $4$ \( 13294 = 2 \cdot 17^{2} \cdot 23 \)
$11$ split multiplicative $12$ \( 39882 = 2 \cdot 3 \cdot 17^{2} \cdot 23 \)
$17$ additive $162$ \( 1518 = 2 \cdot 3 \cdot 11 \cdot 23 \)
$23$ nonsplit multiplicative $24$ \( 19074 = 2 \cdot 3 \cdot 11 \cdot 17^{2} \)

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 438702.l consists of 2 curves linked by isogenies of degree 2.

Twists

The minimal quadratic twist of this elliptic curve is 25806.c2, its twist by $17$.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

$p$-adic regulators

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.