Properties

Label 418950.a1
Conductor $418950$
Discriminant $-1.914\times 10^{14}$
j-invariant \( -\frac{3564664705}{13718} \)
CM no
Rank $0$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3-x^2-76617x-8170709\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-x^2z-76617xz^2-8170709z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-1225875x-524151250\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, -1, 0, -76617, -8170709])
 
gp: E = ellinit([1, -1, 0, -76617, -8170709])
 
magma: E := EllipticCurve([1, -1, 0, -76617, -8170709]);
 
oscar: E = EllipticCurve([1, -1, 0, -76617, -8170709])
 
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: \( 418950 \)  =  $2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 19$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-191414327343750 $  =  $-1 \cdot 2 \cdot 3^{6} \cdot 5^{8} \cdot 7^{2} \cdot 19^{3} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( -\frac{3564664705}{13718} \)  =  $-1 \cdot 2^{-1} \cdot 5 \cdot 7 \cdot 19^{-3} \cdot 467^{3}$
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: $1.5990441720266933760123516739\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $-0.34753893877264727027000262395\dots$
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: $0.14344584479359692027981104131\dots$
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: $9$ = $3^2$ ( exact)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Special value: $ L(E,1) $ ≈ $ 1.2910126031423722825182993718 $
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

$\displaystyle 1.291012603 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{9 \cdot 0.143446 \cdot 1.000000 \cdot 1}{1^2} \approx 1.291012603$

# 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 418950.2.a.a

\( q - q^{2} + q^{4} - q^{8} - 6 q^{11} - 5 q^{13} + q^{16} + 3 q^{17} - q^{19} + O(q^{20}) \) 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: 3207600
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 of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$3$ $1$ $I_0^{*}$ Additive -1 2 6 0
$5$ $1$ $IV^{*}$ Additive -1 2 8 0
$7$ $1$ $II$ Additive -1 2 2 0
$19$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 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], [1726, 1461, 1985, 1589], [3189, 3190, 3182, 3185], [1597, 6, 1599, 19], [3187, 6, 3186, 7], [799, 6, 0, 1], [2739, 2, 2746, 7], [1, 6, 0, 1], [3, 4, 8, 11], [1009, 6, 3027, 19], [1, 0, 6, 1]]
 
GL(2,Integers(3192)).subgroup(gens)
 
Gens := [[4, 3, 9, 7], [1726, 1461, 1985, 1589], [3189, 3190, 3182, 3185], [1597, 6, 1599, 19], [3187, 6, 3186, 7], [799, 6, 0, 1], [2739, 2, 2746, 7], [1, 6, 0, 1], [3, 4, 8, 11], [1009, 6, 3027, 19], [1, 0, 6, 1]];
 
sub<GL(2,Integers(3192))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \), index $16$, genus $0$, and generators

$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1726 & 1461 \\ 1985 & 1589 \end{array}\right),\left(\begin{array}{rr} 3189 & 3190 \\ 3182 & 3185 \end{array}\right),\left(\begin{array}{rr} 1597 & 6 \\ 1599 & 19 \end{array}\right),\left(\begin{array}{rr} 3187 & 6 \\ 3186 & 7 \end{array}\right),\left(\begin{array}{rr} 799 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2739 & 2 \\ 2746 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1009 & 6 \\ 3027 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.

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

The torsion field $K:=\Q(E[3192])$ is a degree-$1143751311360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3192\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 418950.a consists of 2 curves linked by isogenies of degree 3.

Twists

The minimal quadratic twist of this elliptic curve is 46550.bj1, its twist by $-15$.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

$p$-adic regulators

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