# Properties

 Label 110400.gx2 Conductor $110400$ Discriminant $1.203\times 10^{24}$ j-invariant $$\frac{1905890658841300321}{293666194803750}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -41328033, -87610415937]) # or

sage: E = EllipticCurve("110400.gx2")

gp: E = ellinit([0, 1, 0, -41328033, -87610415937]) \\ or

gp: E = ellinit("110400.gx2")

magma: E := EllipticCurve([0, 1, 0, -41328033, -87610415937]); // or

magma: E := EllipticCurve("110400.gx2");

$$y^2=x^3+x^2-41328033x-87610415937$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(10403, 780000\right)$$ $$\hat{h}(P)$$ ≈ $4.0752182253382508717450387594$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-4807, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-4807, 0\right)$$, $$(10403,\pm 780000)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$110400$$ = $$2^{6} \cdot 3 \cdot 5^{2} \cdot 23$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$1202856733916160000000000$$ = $$2^{19} \cdot 3 \cdot 5^{10} \cdot 23^{8}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{1905890658841300321}{293666194803750}$$ = $$2^{-1} \cdot 3^{-1} \cdot 5^{-4} \cdot 23^{-8} \cdot 457^{3} \cdot 2713^{3}$$ 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: $$4.0752182253382508717450387594$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.060160970853034381242010441307$$ 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: $$128$$  = $$2^{2}\cdot1\cdot2^{2}\cdot2^{3}$$ 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

Modular form 110400.2.a.gx

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);

$$q + q^{3} + q^{9} - 4q^{11} - 2q^{13} + 6q^{17} - 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 14155776 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$7.8454107159785282504033866748628547768$$

## 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_9^{*}$$ Additive 1 6 19 1
$$3$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$5$$ $$4$$ $$I_4^{*}$$ Additive 1 2 10 4
$$23$$ $$8$$ $$I_{8}$$ Split multiplicative -1 1 8 8

## 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 X85.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 3 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 3 \end{array}\right)$ and has index 24.

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(3,20) 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 split add ss ordinary ordinary ordinary ordinary split ordinary ss ordinary ordinary ordinary ss - 2 - 1,1 1 1 1 1 2 1 1,1 1 1 1 1,1 - 0 - 0,0 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, 4 and 8.
Its isogeny class 110400.gx consists of 4 curves linked by isogenies of degrees dividing 8.

## 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 $K$ $2$ $$\Q(\sqrt{6})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{-15})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-10})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{6}, \sqrt{-10})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{-5})$$ $$\Z/8\Z$$ Not in database $4$ $$\Q(\sqrt{3}, \sqrt{-10})$$ $$\Z/8\Z$$ Not in database $8$ 8.4.7644119040000.3 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.1911029760000.89 $$\Z/8\Z$$ Not in database $8$ 8.0.3317760000.6 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ 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.