Properties

Label 1050.r2
Conductor 1050
Discriminant -3267280800
j-invariant \( \frac{46969655}{130691232} \)
CM no
Rank 0
Torsion Structure \(\Z/{5}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, 22, -2748]); // or
magma: E := EllipticCurve("1050o1");
sage: E = EllipticCurve([1, 0, 0, 22, -2748]) # or
sage: E = EllipticCurve("1050o1")
gp: E = ellinit([1, 0, 0, 22, -2748]) \\ or
gp: E = ellinit("1050o1")

\( y^2 + x y = x^{3} + 22 x - 2748 \)

Mordell-Weil group structure

\(\Z/{5}\Z\)

Torsion generators

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

\( \left(16, 34\right) \)

Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

\( \left(16, 34\right) \), \( \left(58, 412\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 1050 \)  =  \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(-3267280800 \)  =  \(-1 \cdot 2^{5} \cdot 3^{5} \cdot 5^{2} \cdot 7^{5} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{46969655}{130691232} \)  =  \(2^{-5} \cdot 3^{-5} \cdot 5 \cdot 7^{-5} \cdot 211^{3}\)
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: \(0.656292017735\)
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: \( 125 \)  = \( 5\cdot5\cdot1\cdot5 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
Torsion order: \(5\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 1050.2.a.r

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)

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

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
600 . This curve is \( \Gamma_0(N) \)-optimal.

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])

\( L(E,1) \) ≈ \( 3.28146008867 \)

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\((j)_{-}\)
\(2\) \(5\) \( I_{5} \) Split multiplicative -1 1 5 5
\(3\) \(5\) \( I_{5} \) Split multiplicative -1 1 5 5
\(5\) \(1\) \( II \) Additive 1 2 2 0
\(7\) \(5\) \( I_{5} \) Split multiplicative -1 1 5 5

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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
\(5\) B.1.1

$p$-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

$p$ 2 3 5 7
Reduction type split split add split
$\lambda$-invariant(s) 1 1 - 1
$\mu$-invariant(s) 0 0 - 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ 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 \(d=\) 5.
Its isogeny class 1050.r consists of 2 curves linked by isogenies of degree 5.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.4200.1 \(\Z/10\Z\) Not in database
6 6.0.2963520000.1 \(\Z/2\Z \times \Z/10\Z\) 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.