Properties

Label 110.b2
Conductor 110
Discriminant -1100000
j-invariant \( \frac{109902239}{1100000} \)
CM no
Rank 0
Torsion Structure \(\Z/{5}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, 10, -45]); // or
magma: E := EllipticCurve("110a1");
sage: E = EllipticCurve([1, 1, 1, 10, -45]) # or
sage: E = EllipticCurve("110a1")
gp: E = ellinit([1, 1, 1, 10, -45]) \\ or
gp: E = ellinit("110a1")

\( y^2 + x y + y = x^{3} + x^{2} + 10 x - 45 \)

Mordell-Weil group structure

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

Torsion generators

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

\( \left(3, 3\right) \)

Integral points

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

\( \left(3, 3\right) \), \( \left(13, 43\right) \)

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 110 \)  =  \(2 \cdot 5 \cdot 11\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(-1100000 \)  =  \(-1 \cdot 2^{5} \cdot 5^{5} \cdot 11 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{109902239}{1100000} \)  =  \(2^{-5} \cdot 5^{-5} \cdot 11^{-1} \cdot 479^{3}\)
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: \(1.35954181471\)
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: \( 25 \)  = \( 5\cdot5\cdot1 \)
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 110.2.a.b

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)

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

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 20
\( \Gamma_0(N) \)-optimal: yes
Manin constant: 1

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])

\( L(E,1) \) ≈ \( 1.35954181471 \)

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
\(5\) \(5\) \( I_{5} \) Split multiplicative -1 1 5 5
\(11\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1

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 11
Reduction type split ordinary split split
$\lambda$-invariant(s) 2 0 1 1
$\mu$-invariant(s) 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 110.b 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 3.1.440.1 \(\Z/10\Z\) Not in database
6 6.0.85184000.1 \(\Z/2\Z \times \Z/10\Z\) 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.