Properties

 Label 123.a1 Conductor 123 Discriminant -9963 j-invariant $$-\frac{122023936}{9963}$$ CM no Rank 1 Torsion Structure $$\Z/{5}\Z$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 1, -10, 10]); // or
magma: E := EllipticCurve("123a1");
sage: E = EllipticCurve([0, 1, 1, -10, 10]) # or
sage: E = EllipticCurve("123a1")
gp: E = ellinit([0, 1, 1, -10, 10]) \\ or
gp: E = ellinit("123a1")

$$y^2 + y = x^{3} + x^{2} - 10 x + 10$$

Mordell-Weil group structure

$$\Z\times \Z/{5}\Z$$

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(-4, 1\right)$$ $$\hat{h}(P)$$ ≈ 0.840521417531

Torsion generators

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

$$\left(-1, 4\right)$$

Integral points

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

$$\left(-4, 1\right)$$, $$\left(-1, 4\right)$$, $$\left(1, 1\right)$$, $$\left(2, 1\right)$$, $$\left(5, 10\right)$$, $$\left(8, 22\right)$$, $$\left(107, 1111\right)$$

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

Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$123$$ = $$3 \cdot 41$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$-9963$$ = $$-1 \cdot 3^{5} \cdot 41$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$-\frac{122023936}{9963}$$ = $$-1 \cdot 2^{12} \cdot 3^{-5} \cdot 31^{3} \cdot 41^{-1}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$1$$ magma: Regulator(E); sage: E.regulator() Regulator: $$0.840521417531$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$3.99508269357$$ 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: $$5$$  = $$5\cdot1$$ 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 form123.2.a.a

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)

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

 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])

$$L'(E,1)$$ ≈ $$0.671590513751$$

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$$(j)_{-}$$
$$3$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$41$$ $$1$$ $$I_{1}$$ 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]

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ of good ordinary reduction.

Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 ss split ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary split ordinary ordinary 4,1 2 1 3 1 1 1 1,1 1 1 1 1 2 1 1 0,0 0 0 0 0 0 0 0,0 0 0 0 0 0 0 0

Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 123.a 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 3.1.492.1 $$\Z/10\Z$$ Not in database
6 6.0.29773872.1 $$\Z/2\Z \times \Z/10\Z$$ 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.