# Properties

 Label 5610.q1 Conductor 5610 Discriminant 552431869440 j-invariant $$\frac{901247067798311192691198986281}{552431869440}$$ CM no Rank 0 Torsion Structure $$\Z/{2}\Z$$

# Learn more about

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, -201236483, -1098790303234]); // or
magma: E := EllipticCurve("5610t7");
sage: E = EllipticCurve([1, 0, 1, -201236483, -1098790303234]) # or
sage: E = EllipticCurve("5610t7")
gp: E = ellinit([1, 0, 1, -201236483, -1098790303234]) \\ or
gp: E = ellinit("5610t7")

$$y^2 + x y + y = x^{3} - 201236483 x - 1098790303234$$

## Mordell-Weil group structure

$$\Z/{2}\Z$$

## Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

$$\left(-\frac{32761}{4}, \frac{32757}{8}\right)$$

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()
None

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$5610$$ = $$2 \cdot 3 \cdot 5 \cdot 11 \cdot 17$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$552431869440$$ = $$2^{9} \cdot 3 \cdot 5 \cdot 11^{4} \cdot 17^{3}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{901247067798311192691198986281}{552431869440}$$ = $$2^{-9} \cdot 3^{-1} \cdot 5^{-1} \cdot 11^{-4} \cdot 17^{-3} \cdot 181^{3} \cdot 1163^{3} \cdot 45887^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$0$$ magma: Regulator(E); sage: E.regulator() Regulator: $$1$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$0.0400841681498$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$4$$  = $$1\cdot1\cdot1\cdot2^{2}\cdot1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$36$$ (exact)

## Modular invariants

#### Modular form5610.2.a.q

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

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

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 663552 $$\Gamma_0(N)$$-optimal: no Manin constant: not computed

#### Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar[2]/factorial(ar[1])

$$L(E,1)$$ ≈ $$1.44303005339$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$1$$ $$I_{9}$$ Non-split multiplicative 1 1 9 9
$$3$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$5$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$11$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$17$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X13.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 6.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]

The mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$2$$ B
$$3$$ B.1.2

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

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 1 1 0 0 0

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

## Isogenies

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 up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{510})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$$\Q(\sqrt{-6})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{-85})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{-3})$$ $$\Z/6\Z$$ Not in database
3 3.1.735075.4 $$\Z/6\Z$$ Not in database
4 $$\Q(\sqrt{-6}, \sqrt{-85})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{-3}, \sqrt{-85})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{2}, \sqrt{-3})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{-3}, \sqrt{-170})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
6 $$x^{6}$$ $$\mathstrut +\mathstrut 255 x^{4}$$ $$\mathstrut -\mathstrut 330 x^{3}$$ $$\mathstrut +\mathstrut 21675 x^{2}$$ $$\mathstrut +\mathstrut 84150 x$$ $$\mathstrut +\mathstrut 641350$$ $$\Z/12\Z$$ Not in database
6.0.1621005766875.1 $$\Z/3\Z \times \Z/6\Z$$ Not in database
$$x^{6}$$ $$\mathstrut +\mathstrut 18 x^{4}$$ $$\mathstrut -\mathstrut 330 x^{3}$$ $$\mathstrut +\mathstrut 108 x^{2}$$ $$\mathstrut +\mathstrut 5940 x$$ $$\mathstrut +\mathstrut 27441$$ $$\Z/12\Z$$ Not in database
$$x^{6}$$ $$\mathstrut -\mathstrut 1530 x^{4}$$ $$\mathstrut -\mathstrut 330 x^{3}$$ $$\mathstrut +\mathstrut 780300 x^{2}$$ $$\mathstrut -\mathstrut 504900 x$$ $$\mathstrut -\mathstrut 132623775$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.