# Properties

 Label 82110.bs5 Conductor $82110$ Discriminant $3.636\times 10^{23}$ j-invariant $$\frac{13350979617415439280823624321}{363628103290905600000000}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \oplus \Z/{8}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy=x^3-49423080x+130545230400$$ y^2+xy=x^3-49423080x+130545230400 (homogenize, simplify) $$y^2z+xyz=x^3-49423080xz^2+130545230400z^3$$ y^2z+xyz=x^3-49423080xz^2+130545230400z^3 (dehomogenize, simplify) $$y^2=x^3-64052311707x+6090910426477494$$ y^2=x^3-64052311707x+6090910426477494 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 0, 0, -49423080, 130545230400])

gp: E = ellinit([1, 0, 0, -49423080, 130545230400])

magma: E := EllipticCurve([1, 0, 0, -49423080, 130545230400]);

oscar: E = EllipticCurve([1, 0, 0, -49423080, 130545230400])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z \oplus \Z/{8}\Z$$

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generator and height

 $P$ = $$\left(3480, 24720\right)$$ (3480, 24720) $\hat{h}(P)$ ≈ $2.0348652421815936965216376253$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(3536, -1768\right)$$, $$\left(3060, 87720\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(-7440, 297720\right)$$, $$\left(-7440, -290280\right)$$, $$\left(-4080, 516120\right)$$, $$\left(-4080, -512040\right)$$, $$\left(-1938, 468996\right)$$, $$\left(-1938, -467058\right)$$, $$\left(-510, 394740\right)$$, $$\left(-510, -394230\right)$$, $$\left(960, 289320\right)$$, $$\left(960, -290280\right)$$, $$\left(3060, 87720\right)$$, $$\left(3060, -90780\right)$$, $$\left(3480, 24720\right)$$, $$\left(3480, -28200\right)$$, $$\left(3536, -1768\right)$$, $$\left(4560, -2280\right)$$, $$\left(4760, 53720\right)$$, $$\left(4760, -58480\right)$$, $$\left(5100, 103020\right)$$, $$\left(5100, -108120\right)$$, $$\left(5712, 183192\right)$$, $$\left(5712, -188904\right)$$, $$\left(6480, 283800\right)$$, $$\left(6480, -290280\right)$$, $$\left(8160, 516120\right)$$, $$\left(8160, -524280\right)$$, $$\left(11310, 1003470\right)$$, $$\left(11310, -1014780\right)$$, $$\left(15810, 1808970\right)$$, $$\left(15810, -1824780\right)$$, $$\left(28560, 4677720\right)$$, $$\left(28560, -4706280\right)$$, $$\left(69360, 18141720\right)$$, $$\left(69360, -18211080\right)$$, $$\left(216240, 100394520\right)$$, $$\left(216240, -100610760\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$82110$$ = $2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 23$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $363628103290905600000000$ = $2^{16} \cdot 3^{8} \cdot 5^{8} \cdot 7^{2} \cdot 17^{4} \cdot 23^{2}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{13350979617415439280823624321}{363628103290905600000000}$$ = $2^{-16} \cdot 3^{-8} \cdot 5^{-8} \cdot 7^{-2} \cdot 17^{-4} \cdot 23^{-2} \cdot 241^{3} \cdot 9843601^{3}$ 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: $3.3009530286292500950151736916\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $3.3009530286292500950151736916\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $0.9923666812811367\dots$ Szpiro ratio: $5.723086384275005\dots$

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $2.0348652421815936965216376253\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.095224619621214896073847388293\dots$ 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: $16384$  = $2^{4}\cdot2^{3}\cdot2^{3}\cdot2\cdot2^{2}\cdot2$ 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: $16$ 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: $1$ ( rounded) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L'(E,1)$ ≈ $12.401233194699109664652645957$ 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

$\displaystyle 12.401233195 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.095225 \cdot 2.034865 \cdot 16384}{16^2} \approx 12.401233195$

# 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 82110.2.a.bs

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

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: 14680064
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 semistable. There are 6 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $16$ $I_{16}$ Split multiplicative -1 1 16 16
$3$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$5$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$7$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$17$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$23$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

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
$2$ 2Cs 8.96.0.40

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 = [[199921, 16, 542646, 97], [15, 16, 74, 574849], [1, 16, 0, 1], [618249, 16, 68, 121], [437921, 8, 0, 1], [469209, 16, 187874, 345], [1, 16, 4, 65], [1, 0, 16, 1], [5, 16, 4, 492673], [131377, 8, 0, 1], [656865, 16, 656864, 17]]

GL(2,Integers(656880)).subgroup(gens)

Gens := [[199921, 16, 542646, 97], [15, 16, 74, 574849], [1, 16, 0, 1], [618249, 16, 68, 121], [437921, 8, 0, 1], [469209, 16, 187874, 345], [1, 16, 4, 65], [1, 0, 16, 1], [5, 16, 4, 492673], [131377, 8, 0, 1], [656865, 16, 656864, 17]];

sub<GL(2,Integers(656880))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$656880 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 23$$, index $768$, genus $13$, and generators

$\left(\begin{array}{rr} 199921 & 16 \\ 542646 & 97 \end{array}\right),\left(\begin{array}{rr} 15 & 16 \\ 74 & 574849 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 618249 & 16 \\ 68 & 121 \end{array}\right),\left(\begin{array}{rr} 437921 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 469209 & 16 \\ 187874 & 345 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 16 \\ 4 & 492673 \end{array}\right),\left(\begin{array}{rr} 131377 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 656865 & 16 \\ 656864 & 17 \end{array}\right)$.

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

The torsion field $K:=\Q(E[656880])$ is a degree-$31107765182178263040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/656880\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 82110.bs consists of 8 curves linked by isogenies of degrees dividing 16.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{8}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $4$ $$\Q(i, \sqrt{161})$$ $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $4$ $$\Q(\sqrt{-15}, \sqrt{-119})$$ $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $4$ $$\Q(\sqrt{15}, \sqrt{391})$$ $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $8$ deg 8 $$\Z/2\Z \oplus \Z/24\Z$$ Not in database $16$ deg 16 $$\Z/8\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/16\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/32\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/32\Z$$ 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.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 split split split nonsplit ord ord split ord nonsplit ord ss ord ord ord ss 10 2 2 1 1 1 2 1 1 1 1,1 1 1 1 1,1 1 0 0 0 0 0 0 0 0 0 0,0 0 0 0 0,0

## $p$-adic regulators

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