# Properties

 Label 663.a4 Conductor $663$ Discriminant $25857$ j-invariant $$\frac{17319700013617}{25857}$$ CM no Rank $1$ Torsion structure $$\Z/{4}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -539, 4592])

gp: E = ellinit([1, 1, 1, -539, 4592])

magma: E := EllipticCurve([1, 1, 1, -539, 4592]);

$$y^2+xy+y=x^3+x^2-539x+4592$$

## Mordell-Weil group structure

$$\Z\times \Z/{4}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(9, 19\right)$$ $$\hat{h}(P)$$ ≈ $1.4879060355554164665543321550$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(14, -1\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-12, 103\right)$$, $$\left(-12, -92\right)$$, $$\left(9, 19\right)$$, $$\left(9, -29\right)$$, $$\left(13, -7\right)$$, $$\left(14, -1\right)$$, $$\left(14, -14\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$663$$ = $$3 \cdot 13 \cdot 17$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$25857$$ = $$3^{2} \cdot 13^{2} \cdot 17$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{17319700013617}{25857}$$ = $$3^{-2} \cdot 13^{-2} \cdot 17^{-1} \cdot 25873^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$0.11543517442295990973604623325\dots$$ Stable Faltings height: $$0.11543517442295990973604623325\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1.4879060355554164665543321550\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$3.2047641001061376494479415468\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$4$$  = $$2\cdot2\cdot1$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$4$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 128 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar[2]/factorial(ar[1])

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

$$L'(E,1)$$ ≈ $$1.1920969617698112754438579084628276344$$

## Local data

This elliptic curve is semistable. There are 3 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$13$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$17$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

## 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 X119d.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 8 & 5 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right)$ and has index 48.

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

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

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

Note: $$p$$-adic regulator data only exists for primes $$p\ge 5$$ of good ordinary reduction.

## 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 ordinary nonsplit ordinary ss ordinary split split ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary 2 1 1 1,1 1 2 2 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 0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 663.a consists of 6 curves linked by isogenies of degrees dividing 8.

## 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 $K$ $2$ $$\Q(\sqrt{17})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{221})$$ $$\Z/8\Z$$ Not in database $2$ $$\Q(\sqrt{13})$$ $$\Z/8\Z$$ Not in database $4$ $$\Q(\sqrt{13}, \sqrt{17})$$ $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.0.14295255491821824.58 $$\Z/4\Z \times \Z/4\Z$$ Not in database $8$ 8.0.500516630784.6 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.8.116507435287321.1 $$\Z/2\Z \times \Z/16\Z$$ Not in database $8$ 8.0.28925637601536.48 $$\Z/16\Z$$ Not in database $8$ 8.2.422574120899307.3 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ 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.