Properties

 Label 1200.m3 Conductor $1200$ Discriminant $-1.991\times 10^{15}$ j-invariant $$-\frac{19465109}{248832}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -11208, -2198412])

gp: E = ellinit([0, 1, 0, -11208, -2198412])

magma: E := EllipticCurve([0, 1, 0, -11208, -2198412]);

$$y^2=x^3+x^2-11208x-2198412$$

Mordell-Weil group structure

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

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(158, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(158, 0\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$1200$$ = $$2^{4} \cdot 3 \cdot 5^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-1990656000000000$$ = $$-1 \cdot 2^{22} \cdot 3^{5} \cdot 5^{9}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{19465109}{248832}$$ = $$-1 \cdot 2^{-10} \cdot 3^{-5} \cdot 269^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.6173139103799148336003324588\dots$$ Stable Faltings height: $$-0.28291170450560575676746916258\dots$$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.19905473166115624780264278952\dots$$ 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: $$40$$  = $$2^{2}\cdot5\cdot2$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$2$$ 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);

$$q + q^{3} - 2q^{7} + q^{9} - 2q^{11} + 6q^{13} + 2q^{17} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 4800 $$\Gamma_0(N)$$-optimal: no 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);

$$L(E,1)$$ ≈ $$1.9905473166115624780264278951904589130$$

Local data

This elliptic curve is not semistable. There are 3 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_{14}^{*}$$ Additive -1 4 22 10
$$3$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$5$$ $$2$$ $$III^{*}$$ Additive -1 2 9 0

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 X6.

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

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
$$2$$ B
$$5$$ B.4.1

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

All $$p$$-adic regulators are identically $$1$$ since the rank is $$0$$.

Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) 2 3 5 add split add - 1 - - 0 -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 5 and 10.
Its isogeny class 1200.m consists of 4 curves linked by isogenies of degrees dividing 10.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-15})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{-5})$$ $$\Z/10\Z$$ Not in database $4$ 4.2.96000.1 $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{3}, \sqrt{-5})$$ $$\Z/2\Z \times \Z/10\Z$$ Not in database $8$ 8.0.2916000000.8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.82944000000.8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.2.2834352000000.11 $$\Z/6\Z$$ Not in database $8$ 8.0.9216000000.6 $$\Z/20\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/20\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/20\Z$$ Not in database $16$ Deg 16 $$\Z/30\Z$$ Not in database $20$ 20.4.4882812500000000000000000000.1 $$\Z/10\Z$$ 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.