# Properties

 Label 14400t2 Conductor $14400$ Discriminant $221184000$ j-invariant $$1000188$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -1260, 17200])

gp: E = ellinit([0, 0, 0, -1260, 17200])

magma: E := EllipticCurve([0, 0, 0, -1260, 17200]);

$$y^2=x^3-1260x+17200$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(30, 80\right)$$ $$\left(-34, 144\right)$$ $$\hat{h}(P)$$ ≈ $0.51499726069908523270335455269$ $1.7514652041481706215440208442$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(20, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-40,\pm 60)$$, $$(-34,\pm 144)$$, $$(5,\pm 105)$$, $$(14,\pm 48)$$, $$\left(20, 0\right)$$, $$(21,\pm 1)$$, $$(24,\pm 28)$$, $$(30,\pm 80)$$, $$(36,\pm 136)$$, $$(270,\pm 4400)$$, $$(1230,\pm 43120)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$14400$$ = $$2^{6} \cdot 3^{2} \cdot 5^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$221184000$$ = $$2^{16} \cdot 3^{3} \cdot 5^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$1000188$$ = $$2^{2} \cdot 3^{6} \cdot 7^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$0.52064262099335556358083797907\dots$$ Stable Faltings height: $$-1.0805661700287906988078059921\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.86557586604066943490466574609\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$1.7619864690183152559858781429\dots$$ 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: $$16$$  = $$2^{2}\cdot2\cdot2$$ 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: $$1$$ (rounded)

## Modular invariants

Modular form 14400.2.a.e

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);

$$q - 4q^{7} - 4q^{11} - 4q^{13} - 6q^{17} + 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 8192 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$6.1005318554898775662625215120731278768$$

## 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_6^{*}$$ Additive 1 6 16 0
$$3$$ $$2$$ $$III$$ Additive 1 2 3 0
$$5$$ $$2$$ $$III$$ Additive -1 2 3 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 X47.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 2 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 3 \\ 4 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 3 \\ 2 & 3 \end{array}\right)$ and has index 12.

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

## $p$-adic data

### $p$-adic regulators

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

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

## 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 add add add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary ss ordinary - - - 2 2 2 2 2 2 2 2,2 2 2 2,2 2 - - - 0 0 0 0 0 0 0 0,0 0 0 0,0 0

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.
Its isogeny class 14400t consists of 2 curves linked by isogenies of degree 2.

## 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 $4$ 4.0.216000.2 $$\Z/4\Z$$ Not in database $8$ 8.0.746496000000.7 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.4.746496000000.32 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.2.2239488000000.11 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/4\Z$$ 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

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.