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Properties

Label 441.a2
Conductor $441$
Discriminant $-107163$
j-invariant \( -\frac{28672}{3} \)
CM no
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+y=x^3-21x+40\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+yz^2=x^3-21xz^2+40z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-336x+2576\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, 0, 1, -21, 40])
 
gp: E = ellinit([0, 0, 1, -21, 40])
 
magma: E := EllipticCurve([0, 0, 1, -21, 40]);
 
oscar: E = EllipticCurve([0, 0, 1, -21, 40])
 
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

$P$ =  \(\left(4, 4\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.076270034394571012613674763318$

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

Integral points

\( \left(-5, 4\right) \), \( \left(-5, -5\right) \), \( \left(1, 4\right) \), \( \left(1, -5\right) \), \( \left(2, 2\right) \), \( \left(2, -3\right) \), \( \left(4, 4\right) \), \( \left(4, -5\right) \), \( \left(46, 310\right) \), \( \left(46, -311\right) \) Copy content Toggle raw display

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

Invariants

Conductor: \( 441 \)  =  $3^{2} \cdot 7^{2}$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-107163 $  =  $-1 \cdot 3^{7} \cdot 7^{2} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( -\frac{28672}{3} \)  =  $-1 \cdot 2^{12} \cdot 3^{-1} \cdot 7$
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: $-0.29578137208499745632736253347\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $-1.1694058745949378528758772758\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 

BSD invariants

Analytic rank: $1$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $0.076270034394571012613674763318\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $3.2597903746182327614046723214\dots$
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^{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) $ ≈ $ 0.99449729596489655501469818657 $
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

$\displaystyle 0.994497296 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 3.259790 \cdot 0.076270 \cdot 4}{1^2} \approx 0.994497296$

# 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   441.2.a.a

\( q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{10} + 2 q^{11} - q^{13} - 4 q^{16} - q^{19} + O(q^{20}) \) 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: 48
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$3$ $4$ $I_{1}^{*}$ Additive -1 2 7 1
$7$ $1$ $II$ Additive -1 2 2 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
$13$ 13B.4.1 13.28.0.1

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 = [[335, 26, 78, 251], [14, 23, 325, 339], [1, 26, 0, 379], [181, 520, 169, 207], [1, 0, 26, 1], [1, 26, 0, 1], [521, 26, 520, 27]]
 
GL(2,Integers(546)).subgroup(gens)
 
Gens := [[335, 26, 78, 251], [14, 23, 325, 339], [1, 26, 0, 379], [181, 520, 169, 207], [1, 0, 26, 1], [1, 26, 0, 1], [521, 26, 520, 27]];
 
sub<GL(2,Integers(546))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \), index $336$, genus $9$, and generators

$\left(\begin{array}{rr} 335 & 26 \\ 78 & 251 \end{array}\right),\left(\begin{array}{rr} 14 & 23 \\ 325 & 339 \end{array}\right),\left(\begin{array}{rr} 1 & 26 \\ 0 & 379 \end{array}\right),\left(\begin{array}{rr} 181 & 520 \\ 169 & 207 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 26 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 26 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 521 & 26 \\ 520 & 27 \end{array}\right)$.

The torsion field $K:=\Q(E[546])$ is a degree-$45287424$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/546\Z)$.

$p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

Iwasawa invariants

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

An entry ? indicates that the invariants have not yet been computed.

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 $d=$ 13.
Its isogeny class 441.a consists of 2 curves linked by isogenies of degree 13.

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
$3$ 3.1.588.1 \(\Z/2\Z\) Not in database
$6$ 6.0.1037232.1 \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$6$ \(\Q(\zeta_{21})^+\) \(\Z/13\Z\) Not in database
$8$ 8.2.20841167403.2 \(\Z/3\Z\) Not in database
$12$ deg 12 \(\Z/4\Z\) Not in database
$18$ 18.6.382755853108052250624.1 \(\Z/26\Z\) 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.