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Properties

Label 442225cc2
Conductor $442225$
Discriminant $-3.390\times 10^{16}$
j-invariant \( -162677523113838677 \)
CM no
Rank $1$
Torsion structure trivial

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Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3+x^2-3680780555x-85953932640050\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3+x^2z-3680780555xz^2-85953932640050z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-4770291599955x-4010195126880176850\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 1, 0, -3680780555, -85953932640050])
 
gp: E = ellinit([1, 1, 0, -3680780555, -85953932640050])
 
magma: E := EllipticCurve([1, 1, 0, -3680780555, -85953932640050]);
 
oscar: E = EllipticCurve([1, 1, 0, -3680780555, -85953932640050])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z\)

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generator and height

$P$ =  \(\left(\frac{1110594130903805611171704297301095183570465812169482069857215}{1054878635557107110083120939698009257545330510991438404}, \frac{1167838911180479323530699679798670976673508632586197740622145748637186266779679407762395895}{1083437204751333487639729094764908026273083091987160830794235941780684635342795208}\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $134.54357195425654706227822745$

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

Integral points

None

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

Invariants

Conductor: \( 442225 \)  =  $5^{2} \cdot 7^{2} \cdot 19^{2}$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-33901267729335125 $  =  $-1 \cdot 5^{3} \cdot 7^{8} \cdot 19^{6} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( -162677523113838677 \)  =  $-1 \cdot 7 \cdot 137^{3} \cdot 2083^{3}$
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: $3.7156592926182453455578243565\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $0.54380689222295781849955231162\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $1.0734398000116747\dots$
Szpiro ratio: $5.976563121922419\dots$

BSD invariants

Analytic rank: $1$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $134.54357195425654706227822745\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.0096913566778447387352310831527\dots$
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: $ 4 $  = $ 2\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: $1$
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'(E,1) $ ≈ $ 5.2156389780798731709963429627 $
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

$\displaystyle 5.215638978 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.009691 \cdot 134.543572 \cdot 4}{1^2} \approx 5.215638978$

# 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 analyiic 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.cc

\( q + q^{2} - q^{3} - q^{4} - q^{6} - 3 q^{8} - 2 q^{9} + q^{12} + 2 q^{13} - q^{16} - 2 q^{17} - 2 q^{18} + O(q^{20}) \) 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: 81454464
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 3 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$5$ $2$ $III$ Additive -1 2 3 0
$7$ $1$ $IV^{*}$ Additive 1 2 8 0
$19$ $2$ $I_0^{*}$ Additive -1 2 6 0

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
$37$ 37B.8.2 37.114.4.2

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 = [[56981, 41440, 56980, 56981], [1, 20748, 0, 1], [1, 0, 41440, 1], [82993, 19684, 67488, 1], [1, 0, 49210, 1], [1, 49210, 0, 1], [1, 8436, 0, 1], [78737, 0, 0, 59053], [56979, 0, 0, 98419], [20825, 54834, 41477, 73303], [36175, 59052, 30932, 69311], [1, 0, 67488, 1], [20388, 28823, 97717, 32339], [49211, 49210, 49210, 49211], [56241, 0, 0, 14061], [89985, 8436, 89984, 89985], [82955, 82954, 11951, 12655]]
 
GL(2,Integers(98420)).subgroup(gens)
 
Gens := [[56981, 41440, 56980, 56981], [1, 20748, 0, 1], [1, 0, 41440, 1], [82993, 19684, 67488, 1], [1, 0, 49210, 1], [1, 49210, 0, 1], [1, 8436, 0, 1], [78737, 0, 0, 59053], [56979, 0, 0, 98419], [20825, 54834, 41477, 73303], [36175, 59052, 30932, 69311], [1, 0, 67488, 1], [20388, 28823, 97717, 32339], [49211, 49210, 49210, 49211], [56241, 0, 0, 14061], [89985, 8436, 89984, 89985], [82955, 82954, 11951, 12655]];
 
sub<GL(2,Integers(98420))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 98420 = 2^{2} \cdot 5 \cdot 7 \cdot 19 \cdot 37 \), index $2736$, genus $97$, and generators

$\left(\begin{array}{rr} 56981 & 41440 \\ 56980 & 56981 \end{array}\right),\left(\begin{array}{rr} 1 & 20748 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 41440 & 1 \end{array}\right),\left(\begin{array}{rr} 82993 & 19684 \\ 67488 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 49210 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 49210 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8436 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 78737 & 0 \\ 0 & 59053 \end{array}\right),\left(\begin{array}{rr} 56979 & 0 \\ 0 & 98419 \end{array}\right),\left(\begin{array}{rr} 20825 & 54834 \\ 41477 & 73303 \end{array}\right),\left(\begin{array}{rr} 36175 & 59052 \\ 30932 & 69311 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 67488 & 1 \end{array}\right),\left(\begin{array}{rr} 20388 & 28823 \\ 97717 & 32339 \end{array}\right),\left(\begin{array}{rr} 49211 & 49210 \\ 49210 & 49211 \end{array}\right),\left(\begin{array}{rr} 56241 & 0 \\ 0 & 14061 \end{array}\right),\left(\begin{array}{rr} 89985 & 8436 \\ 89984 & 89985 \end{array}\right),\left(\begin{array}{rr} 82955 & 82954 \\ 11951 & 12655 \end{array}\right)$.

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

The torsion field $K:=\Q(E[98420])$ is a degree-$7617383733657600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/98420\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 37.
Its isogeny class 442225cc consists of 2 curves linked by isogenies of degree 37.

Twists

The minimal quadratic twist of this elliptic curve is 1225h2, its twist by $133$.

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.