Properties

 Label 57c1 Conductor $57$ Discriminant $-1121931$ j-invariant $$\frac{841232384}{1121931}$$ CM no Rank $0$ Torsion structure $$\Z/{5}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, 20, -32])

gp: E = ellinit([0, 1, 1, 20, -32])

magma: E := EllipticCurve([0, 1, 1, 20, -32]);

$$y^2+y=x^3+x^2+20x-32$$

Mordell-Weil group structure

$\Z/{5}\Z$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(2, 4\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(2, 4\right)$$, $$\left(2, -5\right)$$, $$\left(11, 40\right)$$, $$\left(11, -41\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$57$$ = $3 \cdot 19$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-1121931$ = $-1 \cdot 3^{10} \cdot 19$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{841232384}{1121931}$$ = $2^{12} \cdot 3^{-10} \cdot 19^{-1} \cdot 59^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.15331170498862169973663623627\dots$ Stable Faltings height: $-0.15331170498862169973663623627\dots$

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: E.omega[1]  magma: RealPeriod(E); Real period: $1.4662739791414199639605373858\dots$ 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: $10$  = $( 2 \cdot 5 )\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $5$ 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: $L(E,1)$ ≈ $0.58650959165656798558421495433617674241$

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);

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

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

Local data

This elliptic curve is 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$(j)_{-}$
$3$ $10$ $I_{10}$ Split multiplicative -1 1 10 10
$19$ $1$ $I_{1}$ 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
$5$ 5B.1.1 5.24.0.1

$p$-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) $\mu$-invariant(s) 2 3 5 19 ss split ordinary nonsplit 1,4 3 2 0 0,0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 5.
Its isogeny class 57c consists of 2 curves linked by isogenies of degree 5.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{5}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.76.1 $$\Z/10\Z$$ Not in database $6$ 6.0.109744.2 $$\Z/2\Z \times \Z/10\Z$$ Not in database $8$ 8.2.23085974187.2 $$\Z/15\Z$$ Not in database $12$ Deg 12 $$\Z/20\Z$$ Not in database $20$ 20.0.8802533373035313955108642578125.1 $$\Z/5\Z \times \Z/5\Z$$ 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.