# Properties

 Label 82110bt5 Conductor $82110$ Discriminant $-4.541\times 10^{27}$ j-invariant $$\frac{4837987390362436347081585367679}{4540848316592979232425603600}$$ CM no Rank $1$ Torsion structure $$\Z/{4}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy=x^3+352354920x-2007529902000$$ y^2+xy=x^3+352354920x-2007529902000 (homogenize, simplify) $$y^2z+xyz=x^3+352354920xz^2-2007529902000z^3$$ y^2z+xyz=x^3+352354920xz^2-2007529902000z^3 (dehomogenize, simplify) $$y^2=x^3+456651976293x-93664685063640906$$ y^2=x^3+456651976293x-93664685063640906 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 0, 0, 352354920, -2007529902000])

gp: E = ellinit([1, 0, 0, 352354920, -2007529902000])

magma: E := EllipticCurve([1, 0, 0, 352354920, -2007529902000]);

oscar: E = elliptic_curve([1, 0, 0, 352354920, -2007529902000])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z \oplus \Z/{4}\Z$$

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generator and height

 $P$ = $$\left(7350, 985950\right)$$ (7350, 985950) $\hat{h}(P)$ ≈ $8.1394609687263747860865505011$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(26160, 4998180\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(7350, 985950\right)$$, $$\left(7350, -993300\right)$$, $$\left(26160, 4998180\right)$$, $$\left(26160, -5024340\right)$$

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: $-4540848316592979232425603600$ = $-1 \cdot 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 17^{16} \cdot 23^{2}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{4837987390362436347081585367679}{4540848316592979232425603600}$$ = $2^{-4} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-2} \cdot 17^{-16} \cdot 23^{-2} \cdot 71039^{3} \cdot 238081^{3}$ 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: $3.9941002091891954044324058131\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $3.9941002091891954044324058131\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.0168811917145575\dots$ Szpiro ratio: $6.243832285107038\dots$

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $8.1394609687263747860865505011\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.023806154905303724018461847073\dots$ 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: $1024$  = $2^{2}\cdot2\cdot2\cdot2\cdot2^{4}\cdot2$ 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: $4$ 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

$\displaystyle 12.401233195 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.023806 \cdot 8.139461 \cdot 1024}{4^2} \approx 12.401233195$

# 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: 58720256
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 $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $v_p(N)$ $v_p(\Delta)$ $v_p(\mathrm{den}(j))$
$2$ $4$ $I_{4}$ split multiplicative -1 1 4 4
$3$ $2$ $I_{2}$ split multiplicative -1 1 2 2
$5$ $2$ $I_{2}$ split multiplicative -1 1 2 2
$7$ $2$ $I_{2}$ nonsplit multiplicative 1 1 2 2
$17$ $16$ $I_{16}$ split multiplicative -1 1 16 16
$23$ $2$ $I_{2}$ nonsplit 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
$2$ 2B 16.96.0.25

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 = [[25, 16, 1312024, 1312649], [1, 0, 32, 1], [1199549, 24, 631806, 2885], [1, 32, 0, 1], [563069, 8, 374334, 1313477], [437945, 32, 441302, 4329], [1236481, 16, 0, 1], [25, 32, 266134, 4329], [25, 16, 1312374, 1148653], [1313729, 32, 1313728, 33], [1, 32, 4, 129], [25, 16, 942879, 81223]]

GL(2,Integers(1313760)).subgroup(gens)

Gens := [[25, 16, 1312024, 1312649], [1, 0, 32, 1], [1199549, 24, 631806, 2885], [1, 32, 0, 1], [563069, 8, 374334, 1313477], [437945, 32, 441302, 4329], [1236481, 16, 0, 1], [25, 32, 266134, 4329], [25, 16, 1312374, 1148653], [1313729, 32, 1313728, 33], [1, 32, 4, 129], [25, 16, 942879, 81223]];

sub<GL(2,Integers(1313760))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$1313760 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 23$$, index $768$, genus $13$, and generators

$\left(\begin{array}{rr} 25 & 16 \\ 1312024 & 1312649 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1199549 & 24 \\ 631806 & 2885 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 563069 & 8 \\ 374334 & 1313477 \end{array}\right),\left(\begin{array}{rr} 437945 & 32 \\ 441302 & 4329 \end{array}\right),\left(\begin{array}{rr} 1236481 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 25 & 32 \\ 266134 & 4329 \end{array}\right),\left(\begin{array}{rr} 25 & 16 \\ 1312374 & 1148653 \end{array}\right),\left(\begin{array}{rr} 1313729 & 32 \\ 1313728 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 4 & 129 \end{array}\right),\left(\begin{array}{rr} 25 & 16 \\ 942879 & 81223 \end{array}\right)$.

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

The torsion field $K:=\Q(E[1313760])$ is a degree-$497724242914852208640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1313760\Z)$.

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
$2$ split multiplicative $4$ $$1$$
$3$ split multiplicative $4$ $$27370 = 2 \cdot 5 \cdot 7 \cdot 17 \cdot 23$$
$5$ split multiplicative $6$ $$16422 = 2 \cdot 3 \cdot 7 \cdot 17 \cdot 23$$
$7$ nonsplit multiplicative $8$ $$11730 = 2 \cdot 3 \cdot 5 \cdot 17 \cdot 23$$
$17$ split multiplicative $18$ $$4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23$$
$23$ nonsplit multiplicative $24$ $$3570 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 17$$

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 82110bt 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-1})$$ $$\Z/2\Z \oplus \Z/4\Z$$ not in database $2$ $$\Q(\sqrt{15})$$ $$\Z/8\Z$$ not in database $2$ $$\Q(\sqrt{-15})$$ $$\Z/8\Z$$ not in database $4$ $$\Q(i, \sqrt{4830})$$ $$\Z/4\Z \oplus \Z/4\Z$$ not in database $4$ $$\Q(i, \sqrt{322})$$ $$\Z/2\Z \oplus \Z/8\Z$$ not in database $4$ $$\Q(i, \sqrt{15})$$ $$\Z/2\Z \oplus \Z/8\Z$$ not in database $4$ 4.2.28566000.1 $$\Z/16\Z$$ not in database $4$ 4.0.10584000.3 $$\Z/16\Z$$ not in database $8$ deg 8 $$\Z/4\Z \oplus \Z/8\Z$$ not in database $8$ 8.0.816016356000000.57 $$\Z/2\Z \oplus \Z/16\Z$$ not in database $8$ 8.0.1792336896000000.154 $$\Z/2\Z \oplus \Z/16\Z$$ not in database $8$ deg 8 $$\Z/12\Z$$ not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/8\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/16\Z$$ not in database $16$ deg 16 $$\Z/32\Z$$ not in database $16$ deg 16 $$\Z/32\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ not in database $16$ deg 16 $$\Z/24\Z$$ not in database $16$ deg 16 $$\Z/24\Z$$ 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 3 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.