Properties

Label 19.a1
Conductor $19$
Discriminant $-19$
j-invariant \( -\frac{50357871050752}{19} \)
CM no
Rank $0$
Torsion structure trivial

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+y=x^3+x^2-769x-8470\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+yz^2=x^3+x^2z-769xz^2-8470z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-997056x-383201712\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, 1, 1, -769, -8470])
 
gp: E = ellinit([0, 1, 1, -769, -8470])
 
magma: E := EllipticCurve([0, 1, 1, -769, -8470]);
 
oscar: E = EllipticCurve([0, 1, 1, -769, -8470])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

trivial

magma: MordellWeilGroup(E);
 

Integral points

None

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

Invariants

Conductor: \( 19 \)  =  $19$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-19 $  =  $-1 \cdot 19 $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( -\frac{50357871050752}{19} \)  =  $-1 \cdot 2^{18} \cdot 19^{-1} \cdot 577^{3}$
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: $0.033439156991317803520094883584\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $0.033439156991317803520094883584\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 

BSD invariants

Analytic rank: $0$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $1$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.45325324449610360357883918706\dots$
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: $ 1 $
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: $1$ ( exact)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Special value: $ L(E,1) $ ≈ $ 0.45325324449610360357883918707 $
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

$\displaystyle 0.453253244 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.453253 \cdot 1.000000 \cdot 1}{1^2} \approx 0.453253244$

# 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   19.2.a.a

\( q - 2 q^{3} - 2 q^{4} + 3 q^{5} - q^{7} + q^{9} + 3 q^{11} + 4 q^{12} - 4 q^{13} - 6 q^{15} + 4 q^{16} - 3 q^{17} + q^{19} + O(q^{20}) \) 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: 3
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 semistable. There is only one prime of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$19$ $1$ $I_{1}$ Split multiplicative -1 1 1 1

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.1.2 27.72.0.2

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 = [[31, 36, 334, 421], [1, 54, 0, 1], [937, 75, 627, 92], [973, 54, 972, 55], [1, 0, 54, 1], [28, 27, 729, 703], [70, 45, 385, 472]]
 
GL(2,Integers(1026)).subgroup(gens)
 
Gens := [[31, 36, 334, 421], [1, 54, 0, 1], [937, 75, 627, 92], [973, 54, 972, 55], [1, 0, 54, 1], [28, 27, 729, 703], [70, 45, 385, 472]];
 
sub<GL(2,Integers(1026))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1026 = 2 \cdot 3^{3} \cdot 19 \), index $1296$, genus $43$, and generators

$\left(\begin{array}{rr} 31 & 36 \\ 334 & 421 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 937 & 75 \\ 627 & 92 \end{array}\right),\left(\begin{array}{rr} 973 & 54 \\ 972 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right),\left(\begin{array}{rr} 70 & 45 \\ 385 & 472 \end{array}\right)$.

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

The torsion field $K:=\Q(E[1026])$ is a degree-$179508960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1026\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3 and 9.
Its isogeny class 19.a consists of 3 curves linked by isogenies of degrees dividing 9.

Twists

This elliptic curve is its own minimal quadratic twist.

Growth of torsion in number fields

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-3}) \) \(\Z/3\Z\) 2.0.3.1-361.2-a1
$3$ 3.1.76.1 \(\Z/2\Z\) Not in database
$3$ 3.1.1083.1 \(\Z/3\Z\) Not in database
$6$ 6.0.109744.2 \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$6$ 6.0.3518667.2 \(\Z/3\Z \oplus \Z/3\Z\) Not in database
$6$ 6.0.2565108243.2 \(\Z/9\Z\) Not in database
$6$ 6.0.7105563.2 \(\Z/9\Z\) Not in database
$6$ 6.0.155952.1 \(\Z/6\Z\) Not in database
$9$ 9.1.1544610364992.1 \(\Z/6\Z\) Not in database
$12$ 12.2.937292452593664.2 \(\Z/4\Z\) Not in database
$12$ 12.0.8779890495744.1 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$18$ 18.0.16877848680315122776257224907.2 \(\Z/3\Z \oplus \Z/9\Z\) Not in database
$18$ 18.0.64417171850299425397321728.2 \(\Z/3\Z \oplus \Z/6\Z\) Not in database
$18$ 18.0.24956532218240038183849403152084992.1 \(\Z/18\Z\) Not in database
$18$ 18.0.191500465912938345959971172352.1 \(\Z/18\Z\) Not in database
$18$ 18.0.45330602413173669724041216.1 \(\Z/2\Z \oplus \Z/6\Z\) Not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

$p$ 2 3 19
Reduction type ss ord split
$\lambda$-invariant(s) 0,3 0 1
$\mu$-invariant(s) 0,0 2 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

$p$-adic regulators

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