Properties

Label 338.e2
Conductor $338$
Discriminant $-3262922884$
j-invariant \( \frac{351}{4} \)
CM no
Rank $0$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3-x^2+137x+2643\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-x^2z+137xz^2+2643z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+2197x+171366\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, -1, 1, 137, 2643])
 
gp: E = ellinit([1, -1, 1, 137, 2643])
 
magma: E := EllipticCurve([1, -1, 1, 137, 2643]);
 
oscar: E = EllipticCurve([1, -1, 1, 137, 2643])
 
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: \( 338 \)  =  $2 \cdot 13^{2}$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-3262922884 $  =  $-1 \cdot 2^{2} \cdot 13^{8} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{351}{4} \)  =  $2^{-2} \cdot 3^{3} \cdot 13$
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: $0.50644676396357913903945726194\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $-1.2035194743441120183295343658\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $1.2727904629543532\dots$
Szpiro ratio: $5.033189450176238\dots$

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: $1.0433863186966237995349286948\dots$
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: $ 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) $ ≈ $ 2.0867726373932475990698573896 $
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

$\displaystyle 2.086772637 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.043386 \cdot 1.000000 \cdot 2}{1^2} \approx 2.086772637$

# 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   338.2.a.e

\( q + q^{2} + q^{4} - q^{5} + 4 q^{7} + q^{8} - 3 q^{9} - q^{10} + 4 q^{11} + 4 q^{14} + q^{16} + 3 q^{17} - 3 q^{18} + O(q^{20}) \) 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: 156
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$(j)_{-}$
$2$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$13$ $1$ $IV^{*}$ Additive 1 2 8 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
$2$ 2G 4.16.0.2
$7$ 7B.2.1 7.16.0.1

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 = [[57, 77, 329, 106], [1, 0, 196, 1], [345, 336, 84, 305], [1, 28, 0, 157], [15, 7, 336, 169], [169, 168, 196, 169], [1, 208, 336, 1], [197, 0, 0, 85]]
 
GL(2,Integers(364)).subgroup(gens)
 
Gens := [[57, 77, 329, 106], [1, 0, 196, 1], [345, 336, 84, 305], [1, 28, 0, 157], [15, 7, 336, 169], [169, 168, 196, 169], [1, 208, 336, 1], [197, 0, 0, 85]];
 
sub<GL(2,Integers(364))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 364 = 2^{2} \cdot 7 \cdot 13 \), index $768$, genus $21$, and generators

$\left(\begin{array}{rr} 57 & 77 \\ 329 & 106 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 196 & 1 \end{array}\right),\left(\begin{array}{rr} 345 & 336 \\ 84 & 305 \end{array}\right),\left(\begin{array}{rr} 1 & 28 \\ 0 & 157 \end{array}\right),\left(\begin{array}{rr} 15 & 7 \\ 336 & 169 \end{array}\right),\left(\begin{array}{rr} 169 & 168 \\ 196 & 169 \end{array}\right),\left(\begin{array}{rr} 1 & 208 \\ 336 & 1 \end{array}\right),\left(\begin{array}{rr} 197 & 0 \\ 0 & 85 \end{array}\right)$.

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

The torsion field $K:=\Q(E[364])$ is a degree-$6604416$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/364\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 7.
Its isogeny class 338.e consists of 2 curves linked by isogenies of degree 7.

Twists

The minimal quadratic twist of this elliptic curve is 338.c2, its twist by $13$.

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
$3$ 3.1.676.1 \(\Z/4\Z\) Not in database
$3$ 3.3.169.1 \(\Z/7\Z\) 3.3.169.1-8.1-a2
$6$ 6.0.1827904.2 \(\Z/4\Z \oplus \Z/4\Z\) Not in database
$8$ 8.2.999406512.1 \(\Z/3\Z\) Not in database
$9$ 9.3.308915776.1 \(\Z/28\Z\) Not in database
$12$ 12.2.13685690504052736.19 \(\Z/8\Z\) Not in database
$18$ 18.0.6107453226347659264.1 \(\Z/4\Z \oplus \Z/28\Z\) Not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

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

$p$-adic regulators

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