Properties

 Label 100011d1 Conductor 100011 Discriminant 2700297 j-invariant $$\frac{36571225840057}{2700297}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, -692, 6941]); // or
magma: E := EllipticCurve("100011d1");
sage: E = EllipticCurve([1, 0, 1, -692, 6941]) # or
sage: E = EllipticCurve("100011d1")
gp: E = ellinit([1, 0, 1, -692, 6941]) \\ or
gp: E = ellinit("100011d1")

$$y^2 + x y + y = x^{3} - 692 x + 6941$$

Mordell-Weil group structure

$$\Z\times \Z/{2}\Z$$

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(\frac{59}{4}, -\frac{45}{8}\right)$$ $$\hat{h}(P)$$ ≈ 2.71037940969

Torsion generators

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

$$\left(15, -8\right)$$

Integral points

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

$$\left(15, -8\right)$$, $$\left(51, 298\right)$$

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

Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$100011$$ = $$3 \cdot 17 \cdot 37 \cdot 53$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$2700297$$ = $$3^{4} \cdot 17 \cdot 37 \cdot 53$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{36571225840057}{2700297}$$ = $$3^{-4} \cdot 17^{-1} \cdot 19^{3} \cdot 37^{-1} \cdot 53^{-1} \cdot 1747^{3}$$ 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: $$2.71037940969$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$2.43320835565$$ 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: $$4$$  = $$2^{2}\cdot1\cdot1\cdot1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form 100011.2.a.d

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)

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 45312 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$6.59491782665$$

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$$(j)_{-}$$
$$3$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$17$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$37$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$53$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X9.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 2 & 1 \end{array}\right)$ and has index 6.

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
$$2$$ B

$p$-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 53 ordinary split ordinary ordinary ordinary ordinary nonsplit ordinary ordinary ordinary ordinary nonsplit ordinary ordinary ordinary split 3 8 1 1 1 1 1 1 1 1 1 1 1 1 1 2 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 $$d=$$ 2.
Its isogeny class 100011d consists of 2 curves linked by isogenies of degree 2.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{33337})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
4 4.0.533392.1 $$\Z/4\Z$$ 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.