Properties

Label 4620n2
Conductor $4620$
Discriminant $-1.856\times 10^{15}$
j-invariant \( \frac{3132137615458304}{7250937873795} \)
CM no
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3+x^2+19355x-1788745\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3+x^2z+19355xz^2-1788745z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+1567728x-1308698316\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, 1, 0, 19355, -1788745])
 
gp: E = ellinit([0, 1, 0, 19355, -1788745])
 
magma: E := EllipticCurve([0, 1, 0, 19355, -1788745]);
 
oscar: E = EllipticCurve([0, 1, 0, 19355, -1788745])
 
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

$P$ =  \(\left(98, 1029\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.71497333241363115268603992149$

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

Integral points

\((98,\pm 1029)\), \((538,\pm 12831)\) Copy content Toggle raw display

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

Invariants

Conductor: \( 4620 \)  =  $2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-1856240095691520 $  =  $-1 \cdot 2^{8} \cdot 3^{3} \cdot 5 \cdot 7^{9} \cdot 11^{3} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{3132137615458304}{7250937873795} \)  =  $2^{16} \cdot 3^{-3} \cdot 5^{-1} \cdot 7^{-9} \cdot 11^{-3} \cdot 19^{3} \cdot 191^{3}$
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: $1.6138139261182281438798205387\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $1.1517158057449312709349991244\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 

BSD invariants

Analytic rank: $1$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $0.71497333241363115268603992149\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.24290770445507263768042167765\dots$
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: $ 27 $  = $ 1\cdot3\cdot1\cdot3^{2}\cdot1 $
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) $ ≈ $ 4.6891583349260954900707290932 $
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

$\displaystyle 4.689158335 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.242908 \cdot 0.714973 \cdot 27}{1^2} \approx 4.689158335$

# 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("anayltic 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   4620.2.a.n

\( q + q^{3} + q^{5} + q^{7} + q^{9} - q^{11} - 4 q^{13} + q^{15} + 3 q^{17} - 7 q^{19} + O(q^{20}) \) 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: 23328
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$(j)_{-}$
$2$ $1$ $IV^{*}$ Additive -1 2 8 0
$3$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$5$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$7$ $9$ $I_{9}$ Split multiplicative -1 1 9 9
$11$ $1$ $I_{3}$ 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.1.2 3.8.0.2

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 = [[4, 3, 9, 7], [2305, 6, 2304, 7], [661, 6, 1983, 19], [1928, 387, 1, 1156], [1387, 6, 1851, 19], [211, 6, 633, 19], [1, 6, 0, 1], [3, 4, 8, 11], [1, 0, 6, 1]]
 
GL(2,Integers(2310)).subgroup(gens)
 
Gens := [[4, 3, 9, 7], [2305, 6, 2304, 7], [661, 6, 1983, 19], [1928, 387, 1, 1156], [1387, 6, 1851, 19], [211, 6, 633, 19], [1, 6, 0, 1], [3, 4, 8, 11], [1, 0, 6, 1]];
 
sub<GL(2,Integers(2310))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $16$, genus $0$, and generators

$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 2305 & 6 \\ 2304 & 7 \end{array}\right),\left(\begin{array}{rr} 661 & 6 \\ 1983 & 19 \end{array}\right),\left(\begin{array}{rr} 1928 & 387 \\ 1 & 1156 \end{array}\right),\left(\begin{array}{rr} 1387 & 6 \\ 1851 & 19 \end{array}\right),\left(\begin{array}{rr} 211 & 6 \\ 633 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.

The torsion field $K:=\Q(E[2310])$ is a degree-$1108980$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2310\Z)$.

$p$-adic regulators

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

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add split split split nonsplit ord ord ord ord ord ord ord ord ord ord
$\lambda$-invariant(s) - 6 2 2 1 1 1 1 3 1 3 1 1 3 1
$\mu$-invariant(s) - 1 0 0 0 0 0 0 0 0 0 0 0 0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 4620n consists of 2 curves linked by isogenies of degree 3.

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
$2$ \(\Q(\sqrt{-3}) \) \(\Z/3\Z\) Not in database
$3$ 3.1.4620.1 \(\Z/2\Z\) Not in database
$3$ 3.1.2700.1 \(\Z/3\Z\) Not in database
$6$ 6.0.24652782000.1 \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$6$ 6.0.21870000.2 \(\Z/3\Z \oplus \Z/3\Z\) Not in database
$6$ 6.0.64033200.1 \(\Z/6\Z\) Not in database
$9$ 9.1.134789085585000000.1 \(\Z/6\Z\) Not in database
$12$ deg 12 \(\Z/4\Z\) Not in database
$12$ deg 12 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$18$ 18.0.38291826458136784924460203200000000.4 \(\Z/9\Z\) Not in database
$18$ 18.0.54504292778521364376675000000000000.1 \(\Z/3\Z \oplus \Z/6\Z\) Not in database
$18$ 18.0.124415041475283470214882838875000000000000.1 \(\Z/2\Z \oplus \Z/6\Z\) Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.