# Properties

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

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3-5x-8$$ y^2+xy+y=x^3-5x-8 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-5xz^2-8z^3$$ y^2z+xyz+yz^2=x^3-5xz^2-8z^3 (dehomogenize, simplify) $$y^2=x^3-5859x-344034$$ y^2=x^3-5859x-344034 (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

$$\left(4, 4\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(4, 4\right)$$, $$\left(4, -9\right)$$

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

$$q - q^{2} + q^{3} + q^{4} - 3 q^{5} - q^{6} - q^{7} - q^{8} - 2 q^{9} + 3 q^{10} + 6 q^{11} + q^{12} + q^{13} + q^{14} - 3 q^{15} + q^{16} - 3 q^{17} + 2 q^{18} + 2 q^{19} + O(q^{20})$$

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);

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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/3\Z \oplus \Z/3\Z$$ 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

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