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Properties

Label 26a1
Conductor $26$
Discriminant $-17576$
j-invariant \( -\frac{10218313}{17576} \)
CM no
Rank $0$
Torsion structure \(\Z/{3}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3-5x-8\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-5xz^2-8z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-5859x-344034\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 0, 1, -5, -8])
 
gp: E = ellinit([1, 0, 1, -5, -8])
 
magma: E := EllipticCurve([1, 0, 1, -5, -8]);
 
oscar: E = elliptic_curve([1, 0, 1, -5, -8])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z/{3}\Z\)

magma: MordellWeilGroup(E);
 

Torsion generators

\( \left(4, 4\right) \) Copy content Toggle raw display

comment: Torsion subgroup
 
sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 
oscar: torsion_structure(E)
 

Integral points

\( \left(4, 4\right) \), \( \left(4, -9\right) \) Copy content Toggle raw display

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

Invariants

Conductor: \( 26 \)  =  $2 \cdot 13$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-17576 $  =  $-1 \cdot 2^{3} \cdot 13^{3} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( -\frac{10218313}{17576} \)  =  $-1 \cdot 2^{-3} \cdot 7^{3} \cdot 13^{-3} \cdot 31^{3}$
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: $-0.49491971820067818992441210777\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $-0.49491971820067818992441210777\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $0.947172997962475\dots$
Szpiro ratio: $5.377073159063485\dots$

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: $1.5467299538318833526303052273\dots$
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: $ 3 $  = $ 1\cdot3 $
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: $3$
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) $ ≈ $ 0.51557665127729445087676840909 $
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

$\displaystyle 0.515576651 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.546730 \cdot 1.000000 \cdot 3}{3^2} \approx 0.515576651$

# 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   26.2.a.a

\( q - q^{2} + q^{3} + q^{4} - 3 q^{5} - q^{6} - q^{7} - q^{8} - 2 q^{9} + 3 q^{10} + 6 q^{11} + q^{12} + q^{13} + q^{14} - 3 q^{15} + q^{16} - 3 q^{17} + 2 q^{18} + 2 q^{19} + O(q^{20}) \) 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: 2
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 semistable. There are 2 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $v_p(N)$ $v_p(\Delta)$ $v_p(\mathrm{den}(j))$
$2$ $1$ $I_{3}$ nonsplit multiplicative 1 1 3 3
$13$ $3$ $I_{3}$ 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
$3$ 3Cs.1.1 3.24.0.1

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 = [[469, 18, 477, 163], [13, 18, 153, 511], [1, 18, 0, 1], [1, 0, 18, 1], [713, 486, 306, 541], [919, 18, 918, 19], [1, 12, 0, 1], [1, 6, 6, 37], [1, 9, 9, 82], [703, 18, 711, 163]]
 
GL(2,Integers(936)).subgroup(gens)
 
Gens := [[469, 18, 477, 163], [13, 18, 153, 511], [1, 18, 0, 1], [1, 0, 18, 1], [713, 486, 306, 541], [919, 18, 918, 19], [1, 12, 0, 1], [1, 6, 6, 37], [1, 9, 9, 82], [703, 18, 711, 163]];
 
sub<GL(2,Integers(936))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \), index $144$, genus $3$, and generators

$\left(\begin{array}{rr} 469 & 18 \\ 477 & 163 \end{array}\right),\left(\begin{array}{rr} 13 & 18 \\ 153 & 511 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 713 & 486 \\ 306 & 541 \end{array}\right),\left(\begin{array}{rr} 919 & 18 \\ 918 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 703 & 18 \\ 711 & 163 \end{array}\right)$.

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

The torsion field $K:=\Q(E[936])$ is a degree-$1086898176$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/936\Z)$.

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$ Reduction type Serre weight Serre conductor
$2$ nonsplit multiplicative $4$ \( 13 \)
$3$ good $2$ \( 1 \)
$13$ split multiplicative $14$ \( 2 \)

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 26a 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-3}) \) \(\Z/3\Z \oplus \Z/3\Z\) 2.0.3.1-676.2-b2
$3$ 3.1.104.1 \(\Z/6\Z\) not in database
$6$ 6.0.1124864.1 \(\Z/2\Z \oplus \Z/6\Z\) not in database
$6$ 6.0.292032.1 \(\Z/3\Z \oplus \Z/6\Z\) not in database
$9$ 9.3.95006081547.1 \(\Z/9\Z\) not in database
$12$ 12.2.8421963387109376.8 \(\Z/12\Z\) not in database
$12$ 12.0.922417564483584.1 \(\Z/6\Z \oplus \Z/6\Z\) not in database
$18$ 18.0.727702811914257020738325491712.1 \(\Z/3\Z \oplus \Z/9\Z\) not in database
$18$ 18.0.243706199334710775656643.1 \(\Z/3\Z \oplus \Z/9\Z\) not in database
$18$ 18.0.150762711330458077114368.1 \(\Z/3\Z \oplus \Z/9\Z\) not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

$p$ 2 3 13
Reduction type nonsplit ord split
$\lambda$-invariant(s) 1 0 1
$\mu$-invariant(s) 0 1 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 $0$.