Properties

 Label 203b1 Conductor $203$ Discriminant $-1421$ j-invariant $$-\frac{1}{1421}$$ CM no Rank $1$ Torsion structure trivial

Related objects

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, 0, -2])

gp: E = ellinit([1, 1, 1, 0, -2])

magma: E := EllipticCurve([1, 1, 1, 0, -2]);

$$y^2+xy+y=x^3+x^2-2$$

Mordell-Weil group structure

$$\Z$$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(2, 2\right)$$ $$\hat{h}(P)$$ ≈ $0.20789228685957103627216072612$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(1, 0\right)$$, $$\left(1, -2\right)$$, $$\left(2, 2\right)$$, $$\left(2, -5\right)$$, $$\left(16, 58\right)$$, $$\left(16, -75\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$203$$ = $$7 \cdot 29$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-1421$$ = $$-1 \cdot 7^{2} \cdot 29$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{1}{1421}$$ = $$-1 \cdot 7^{-2} \cdot 29^{-1}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$-0.71619004926121542193742357951\dots$$ Stable Faltings height: $$-0.71619004926121542193742357951\dots$$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.20789228685957103627216072612\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$2.1960815500483164397202960575\dots$$ 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: $$2$$  = $$2\cdot1$$ 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

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 8 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$0.91309683113931201763607581886994391382$$

Local data

This elliptic curve is semistable. There are 2 primes of bad reduction:

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$$(j)_{-}$$
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$29$$ $$1$$ $$I_{1}$$ 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(5,20) if E.conductor().valuation(p)<2]

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

Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 ordinary ordinary ordinary split ordinary ordinary ordinary ordinary ordinary split ordinary ordinary ss ordinary ordinary 3 7 1 4 3 1 1 1 1 2 1 1 1,1 1 1 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 203b consists of this curve only.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.116.1 $$\Z/2\Z$$ Not in database $6$ 6.0.1560896.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.45850905387.2 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.