Properties

Label 1110j1
Conductor $1110$
Discriminant $-5328000$
j-invariant \( -\frac{243087455521}{5328000} \)
CM no
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -130, 527]) # or
 
sage: E = EllipticCurve("1110j1")
 
gp: E = ellinit([1, 1, 1, -130, 527]) \\ or
 
gp: E = ellinit("1110j1")
 
magma: E := EllipticCurve([1, 1, 1, -130, 527]); // or
 
magma: E := EllipticCurve("1110j1");
 

\( y^2 + x y + y = x^{3} + x^{2} - 130 x + 527 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

\(P\) =  \( \left(17, -69\right) \)
\(\hat{h}(P)\) ≈  $0.029997549311021145$

Integral points

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

\( \left(-13, 21\right) \), \( \left(-13, -9\right) \), \( \left(-3, 31\right) \), \( \left(-3, -29\right) \), \( \left(1, 19\right) \), \( \left(1, -21\right) \), \( \left(5, 3\right) \), \( \left(5, -9\right) \), \( \left(7, 1\right) \), \( \left(7, -9\right) \), \( \left(17, 51\right) \), \( \left(17, -69\right) \), \( \left(35, 183\right) \), \( \left(35, -219\right) \), \( \left(157, 1891\right) \), \( \left(157, -2049\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 1110 \)  =  \(2 \cdot 3 \cdot 5 \cdot 37\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-5328000 \)  =  \(-1 \cdot 2^{7} \cdot 3^{2} \cdot 5^{3} \cdot 37 \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{243087455521}{5328000} \)  =  \(-1 \cdot 2^{-7} \cdot 3^{-2} \cdot 5^{-3} \cdot 37^{-1} \cdot 79^{6}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(0.029997549311\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(2.41451878814\)
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: \( 42 \)  = \( 7\cdot2\cdot3\cdot1 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(1\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form   1110.2.a.j

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 336
\( \Gamma_0(N) \)-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);
 

\( L'(E,1) \) ≈ \( 3.0420451492 \)

Local data

This elliptic curve is semistable.

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\((j)_{-}\)
\(2\) \(7\) \( I_{7} \) Split multiplicative -1 1 7 7
\(3\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(5\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3
\(37\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1

Galois representations

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

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 \( p \) .

$p$-adic data

$p$-adic regulators

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

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

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type split nonsplit split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary nonsplit ordinary ordinary ss
$\lambda$-invariant(s) 11 1 2 1 1 1 1 1 1 1 1 1 1 1 1,1
$\mu$-invariant(s) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0,0

Isogenies

This curve has no rational isogenies. Its isogeny class 1110j consists of this curve only.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.1480.1 \(\Z/2\Z\) Not in database
$6$ 6.0.3241792000.1 \(\Z/2\Z \times \Z/2\Z\) 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.