# Properties

 Label 5610.t3 Conductor $5610$ Discriminant $-1.009\times 10^{16}$ j-invariant $$-\frac{9354997870579612441}{10093752054144000}$$ CM no Rank $0$ Torsion structure $$\Z/{6}\Z$$

# Learn more

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3-43898x+5987756$$ y^2+xy+y=x^3-43898x+5987756 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-43898xz^2+5987756z^3$$ y^2z+xyz+yz^2=x^3-43898xz^2+5987756z^3 (dehomogenize, simplify) $$y^2=x^3-56891187x+279535429134$$ y^2=x^3-56891187x+279535429134 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 0, 1, -43898, 5987756])

gp: E = ellinit([1, 0, 1, -43898, 5987756])

magma: E := EllipticCurve([1, 0, 1, -43898, 5987756]);

oscar: E = EllipticCurve([1, 0, 1, -43898, 5987756])

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

$$\left(285, 3937\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(-259, 129\right)$$, $$\left(30, 2152\right)$$, $$\left(30, -2183\right)$$, $$\left(285, 3937\right)$$, $$\left(285, -4223\right)$$

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: $-10093752054144000$ = $-1 \cdot 2^{10} \cdot 3^{3} \cdot 5^{3} \cdot 11^{2} \cdot 17^{6}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-\frac{9354997870579612441}{10093752054144000}$$ = $-1 \cdot 2^{-10} \cdot 3^{-3} \cdot 5^{-3} \cdot 11^{-2} \cdot 17^{-6} \cdot 19^{3} \cdot 110899^{3}$ 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: $1.7657811263480875411494360145\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $1.7657811263480875411494360145\dots$ 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: $0.36995441689872125734412204100\dots$ 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: $216$  = $2\cdot3\cdot3\cdot2\cdot( 2 \cdot 3 )$ 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: $L(E,1)$ ≈ $2.2197265013923275440647322460$ 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

$\displaystyle 2.219726501 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.369954 \cdot 1.000000 \cdot 216}{6^2} \approx 2.219726501$

# 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

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

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);

For more coefficients, see the Downloads section to the right.

Modular degree: 46080
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 5 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $I_{10}$ Non-split multiplicative 1 1 10 10
$3$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$5$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$11$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$17$ $6$ $I_{6}$ Split multiplicative -1 1 6 6

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 2.3.0.1
$3$ 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, 6, 6, 37], [1, 12, 0, 1], [11, 2, 1990, 2031], [1706, 11, 669, 2020], [10, 3, 381, 2032], [2029, 12, 2028, 13], [241, 12, 1446, 73], [435, 88, 398, 77], [1021, 12, 6, 73], [1, 0, 12, 1]]

GL(2,Integers(2040)).subgroup(gens)

Gens := [[1, 6, 6, 37], [1, 12, 0, 1], [11, 2, 1990, 2031], [1706, 11, 669, 2020], [10, 3, 381, 2032], [2029, 12, 2028, 13], [241, 12, 1446, 73], [435, 88, 398, 77], [1021, 12, 6, 73], [1, 0, 12, 1]];

sub<GL(2,Integers(2040))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17$$, index $96$, genus $1$, and generators

$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 1990 & 2031 \end{array}\right),\left(\begin{array}{rr} 1706 & 11 \\ 669 & 2020 \end{array}\right),\left(\begin{array}{rr} 10 & 3 \\ 381 & 2032 \end{array}\right),\left(\begin{array}{rr} 2029 & 12 \\ 2028 & 13 \end{array}\right),\left(\begin{array}{rr} 241 & 12 \\ 1446 & 73 \end{array}\right),\left(\begin{array}{rr} 435 & 88 \\ 398 & 77 \end{array}\right),\left(\begin{array}{rr} 1021 & 12 \\ 6 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right)$.

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

The torsion field $K:=\Q(E[2040])$ is a degree-$28877783040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2040\Z)$.

## Isogenies

gp: ellisomat(E)

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

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-15})$$ $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $4$ 4.2.277440.2 $$\Z/12\Z$$ Not in database $6$ 6.0.6324912.1 $$\Z/3\Z \oplus \Z/6\Z$$ Not in database $8$ 8.0.17318914560000.3 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $8$ deg 8 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $9$ 9.3.1267719288933920550750000.1 $$\Z/18\Z$$ Not in database $12$ deg 12 $$\Z/6\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/24\Z$$ Not in database $18$ 18.0.5424003659931047334549866229006993723773437500000000.1 $$\Z/2\Z \oplus \Z/18\Z$$ 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 11 17 nonsplit split split nonsplit split 5 7 1 0 1 0 0 0 0 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 $0$.