# Properties

 Label 5610.q5 Conductor $5610$ Discriminant $5.862\times 10^{18}$ j-invariant $$\frac{476646772170172569823801}{5862293314453125000}$$ CM no Rank $0$ Torsion structure $$\Z/{6}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3-1627388x+790397138$$ y^2+xy+y=x^3-1627388x+790397138 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-1627388xz^2+790397138z^3$$ y^2z+xyz+yz^2=x^3-1627388xz^2+790397138z^3 (dehomogenize, simplify) $$y^2=x^3-2109094227x+36883096164846$$ y^2=x^3-2109094227x+36883096164846 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 0, 1, -1627388, 790397138])

gp: E = ellinit([1, 0, 1, -1627388, 790397138])

magma: E := EllipticCurve([1, 0, 1, -1627388, 790397138]);

oscar: E = EllipticCurve([1, 0, 1, -1627388, 790397138])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z/{6}\Z$$

magma: MordellWeilGroup(E);

## Torsion generators

$$\left(2604, 117910\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(954, 9835\right)$$, $$\left(954, -10790\right)$$, $$\left(2604, 117910\right)$$, $$\left(2604, -120515\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$5610$$ = $2 \cdot 3 \cdot 5 \cdot 11 \cdot 17$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $5862293314453125000$ = $2^{3} \cdot 3^{3} \cdot 5^{12} \cdot 11^{3} \cdot 17^{4}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{476646772170172569823801}{5862293314453125000}$$ = $2^{-3} \cdot 3^{-3} \cdot 5^{-12} \cdot 11^{-3} \cdot 17^{-4} \cdot 23^{3} \cdot 107^{3} \cdot 31741^{3}$ 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: $2.4114813045780072331167963565\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $2.4114813045780072331167963565\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## 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: $0.24050500889864283326946327256\dots$ 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: $216$  = $1\cdot3\cdot( 2^{2} \cdot 3 )\cdot3\cdot2$ 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: $6$ 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: $L(E,1)$ ≈ $1.4430300533918569996167796354$ 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

$\displaystyle 1.443030053 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.240505 \cdot 1.000000 \cdot 216}{6^2} \approx 1.443030053$

# 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("anayltic 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

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

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: 221184
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 5 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$3$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$5$ $12$ $I_{12}$ Split multiplicative -1 1 12 12
$11$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$17$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4

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 4.6.0.1
$3$ 3B.1.1 3.8.0.1

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 = [[1, 12, 12, 145], [14521, 24, 17172, 289], [16831, 24, 16830, 1], [15, 106, 21134, 13211], [12162, 15919, 7037, 11972], [2056, 3, 9741, 22354], [1, 24, 0, 1], [22417, 24, 22416, 25], [16, 11241, 3455, 7106], [8977, 24, 17964, 289], [1, 0, 24, 1]]

GL(2,Integers(22440)).subgroup(gens)

Gens := [[1, 12, 12, 145], [14521, 24, 17172, 289], [16831, 24, 16830, 1], [15, 106, 21134, 13211], [12162, 15919, 7037, 11972], [2056, 3, 9741, 22354], [1, 24, 0, 1], [22417, 24, 22416, 25], [16, 11241, 3455, 7106], [8977, 24, 17964, 289], [1, 0, 24, 1]];

sub<GL(2,Integers(22440))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$22440 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 17$$, index $384$, genus $5$, and generators

$\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 14521 & 24 \\ 17172 & 289 \end{array}\right),\left(\begin{array}{rr} 16831 & 24 \\ 16830 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 21134 & 13211 \end{array}\right),\left(\begin{array}{rr} 12162 & 15919 \\ 7037 & 11972 \end{array}\right),\left(\begin{array}{rr} 2056 & 3 \\ 9741 & 22354 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 22417 & 24 \\ 22416 & 25 \end{array}\right),\left(\begin{array}{rr} 16 & 11241 \\ 3455 & 7106 \end{array}\right),\left(\begin{array}{rr} 8977 & 24 \\ 17964 & 289 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.

The torsion field $K:=\Q(E[22440])$ is a degree-$459571220$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/22440\Z)$.

## $p$-adic regulators

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

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 11 17 nonsplit split split split nonsplit 5 3 1 3 0 0 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3, 4, 6 and 12.
Its isogeny class 5610.q consists of 8 curves linked by isogenies of degrees dividing 12.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{66})$$ $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $2$ $$\Q(\sqrt{6})$$ $$\Z/12\Z$$ Not in database $2$ $$\Q(\sqrt{11})$$ $$\Z/12\Z$$ Not in database $4$ $$\Q(\sqrt{6}, \sqrt{11})$$ $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $6$ 6.0.2255067.2 $$\Z/3\Z \oplus \Z/6\Z$$ Not in database $8$ 8.0.5416809268248576.19 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $8$ deg 8 $$\Z/24\Z$$ Not in database $8$ deg 8 $$\Z/24\Z$$ Not in database $9$ 9.3.33173450251082263546875.3 $$\Z/18\Z$$ Not in database $12$ deg 12 $$\Z/6\Z \oplus \Z/6\Z$$ Not in database $12$ deg 12 $$\Z/3\Z \oplus \Z/12\Z$$ Not in database $12$ deg 12 $$\Z/3\Z \oplus \Z/12\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/12\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/24\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/24\Z$$ Not in database $18$ 18.6.5308025359933309041398141791540791330720743424000000000000.1 $$\Z/2\Z \oplus \Z/18\Z$$ Not in database $18$ 18.6.3987998016478819715550820279144095665455104000000000000.1 $$\Z/36\Z$$ Not in database $18$ 18.6.383971741893323860054842432837152150659776000000000000.1 $$\Z/36\Z$$ 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.