Properties

 Label 1155.h2 Conductor $1155$ Discriminant $-509355$ j-invariant $$-\frac{262144}{509355}$$ CM no Rank $1$ Torsion structure $$\Z/{3}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, -1, -35]) # or

sage: E = EllipticCurve("1155.h2")

gp: E = ellinit([0, 1, 1, -1, -35]) \\ or

gp: E = ellinit("1155.h2")

magma: E := EllipticCurve([0, 1, 1, -1, -35]); // or

magma: E := EllipticCurve("1155.h2");

$$y^2+y=x^3+x^2-x-35$$

Mordell-Weil group structure

$$\Z\times \Z/{3}\Z$$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(11, 37\right)$$ $$\hat{h}(P)$$ ≈ $1.8702142681361649772593541023$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(5, 10\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(5, 10\right)$$, $$\left(5, -11\right)$$, $$\left(11, 37\right)$$, $$\left(11, -38\right)$$, $$\left(47, 325\right)$$, $$\left(47, -326\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$1155$$ = $$3 \cdot 5 \cdot 7 \cdot 11$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-509355$$ = $$-1 \cdot 3^{3} \cdot 5 \cdot 7^{3} \cdot 11$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{262144}{509355}$$ = $$-1 \cdot 2^{18} \cdot 3^{-3} \cdot 5^{-1} \cdot 7^{-3} \cdot 11^{-1}$$ 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); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1.8702142681361649772593541023$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$1.3296808296840800339007487124$$ 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: $$9$$  = $$3\cdot1\cdot3\cdot1$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$3$$ 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);

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

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 144 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$2.4867880597423003717912357908630641757$$

Local data

This elliptic curve is semistable. There are 4 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$$(j)_{-}$$
$$3$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$5$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$7$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$11$$ $$1$$ $$I_{1}$$ 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 $$p$$ except those listed.

prime Image of Galois representation
$$3$$ 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 $$p\ge 5$$ 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 ss split nonsplit split nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary 8,1 2 1 2 1 1 1 1 1 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 non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 1155.h consists of 2 curves linked by isogenies of degree 3.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.4620.1 $$\Z/6\Z$$ Not in database $6$ 6.0.24652782000.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $6$ 6.0.247066875.1 $$\Z/3\Z \times \Z/3\Z$$ Not in database $9$ 9.3.15447486304833226875.2 $$\Z/9\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $18$ 18.0.21984527369360325045258675000000000000.1 $$\Z/3\Z \times \Z/6\Z$$ 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.