# Properties

 Label 171.a3 Conductor $171$ Discriminant $41553$ j-invariant $$\frac{389017}{57}$$ CM no Rank $0$ Torsion structure $$\Z/{4}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3-x^2-14x+20$$ y^2+xy+y=x^3-x^2-14x+20 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-x^2z-14xz^2+20z^3$$ y^2z+xyz+yz^2=x^3-x^2z-14xz^2+20z^3 (dehomogenize, simplify) $$y^2=x^3-219x+1078$$ y^2=x^3-219x+1078 (homogenize, minimize)

sage: E = EllipticCurve([1, -1, 1, -14, 20])

gp: E = ellinit([1, -1, 1, -14, 20])

magma: E := EllipticCurve([1, -1, 1, -14, 20]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(0, 4\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(0, 4\right)$$, $$\left(0, -5\right)$$, $$\left(3, -2\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$171$$ = $3^{2} \cdot 19$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $41553$ = $3^{7} \cdot 19$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{389017}{57}$$ = $3^{-1} \cdot 19^{-1} \cdot 73^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.38884969842871655738904187287\dots$ Stable Faltings height: $-0.93815584276277140308666449133\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ 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); Real period: $3.4734453619135780883065631268\dots$ 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: $4$  = $2^{2}\cdot1$ 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) 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); Special value: $L(E,1)$ ≈ $0.86836134047839452207664078170$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 12 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 2 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$(j)_{-}$
$3$ $4$ $I_{1}^{*}$ Additive -1 2 7 1
$19$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

## Galois representations

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 $\ell$-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 4.12.0.7
sage: gens = [[7, 6, 450, 451], [65, 60, 62, 287], [388, 1, 407, 6], [1, 0, 8, 1], [296, 453, 299, 454], [449, 8, 448, 9], [403, 402, 298, 67], [1, 8, 0, 1], [1, 4, 4, 17]]

sage: GL(2,Integers(456)).subgroup(gens)

magma: Gens := [[7, 6, 450, 451], [65, 60, 62, 287], [388, 1, 407, 6], [1, 0, 8, 1], [296, 453, 299, 454], [449, 8, 448, 9], [403, 402, 298, 67], [1, 8, 0, 1], [1, 4, 4, 17]];

magma: sub<GL(2,Integers(456))|Gens>;

The image of the adelic Galois representation has level $456$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 7 & 6 \\ 450 & 451 \end{array}\right),\left(\begin{array}{rr} 65 & 60 \\ 62 & 287 \end{array}\right),\left(\begin{array}{rr} 388 & 1 \\ 407 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 296 & 453 \\ 299 & 454 \end{array}\right),\left(\begin{array}{rr} 449 & 8 \\ 448 & 9 \end{array}\right),\left(\begin{array}{rr} 403 & 402 \\ 298 & 67 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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) 2 3 19 ord add nonsplit 5 - 0 0 - 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

An entry - indicates that the invariants are not computed because the reduction is additive.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 171.a consists of 4 curves linked by isogenies of degrees dividing 4.

## 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 $K$ $2$ $$\Q(\sqrt{57})$$ $$\Z/2\Z \oplus \Z/4\Z$$ 2.2.57.1-57.1-d3 $4$ 4.0.513.1 $$\Z/8\Z$$ Not in database $8$ 8.0.8779890495744.2 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.95004009.1 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.8.975543388416.1 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.2.23085974187.3 $$\Z/12\Z$$ Not in database $16$ deg 16 $$\Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ 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.