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## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -133408, 18675188])

gp: E = ellinit([0, 1, 0, -133408, 18675188])

magma: E := EllipticCurve([0, 1, 0, -133408, 18675188]);

$$y^2=x^3+x^2-133408x+18675188$$ ## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(68, 3150\right)$$ $$\hat{h}(P)$$ ≈ $2.6887061017738908904408359333$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(203, 0\right)$$, $$\left(218, 0\right)$$ ## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-422, 0\right)$$, $$(68,\pm 3150)$$, $$(154,\pm 1344)$$, $$\left(203, 0\right)$$, $$\left(218, 0\right)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$1200$$ = $$2^{4} \cdot 3 \cdot 5^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$576000000000000$$ = $$2^{18} \cdot 3^{2} \cdot 5^{12}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{4102915888729}{9000000}$$ = $$2^{-6} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{3} \cdot 2287^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.7154585317133745222982380489\dots$$ Stable Faltings height: $$0.21759239493637902558062626083\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$2.6887061017738908904408359333\dots$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.51804870109955817783027952757\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$32$$  = $$2^{2}\cdot2\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$4$$ 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)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q + q^{3} - 4q^{7} + q^{9} - 2q^{13} - 6q^{17} + 4q^{19} + O(q^{20})$$ sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 6912 $$\Gamma_0(N)$$-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/factorial(ar)

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

$$L'(E,1)$$ ≈ $$2.7857614073248413034447420955686775507$$

## Local data

This elliptic curve is not semistable. There are 3 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$4$$ $$I_{10}^{*}$$ Additive -1 4 18 6
$$3$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$5$$ $$4$$ $$I_6^{*}$$ Additive 1 2 12 6

## 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 X8d.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 2 \\ 0 & 1 \end{array}\right)$ 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
$$2$$ Cs
$$3$$ 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 split add ordinary ss ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ss - 2 - 1 1,3 1 3 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

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, 3 and 6.
Its isogeny class 1200.k consists of 4 curves linked by isogenies of degrees dividing 12.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-15})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{15})$$ $$\Z/2\Z \times \Z/6\Z$$ 2.2.60.1-30.1-d6 $4$ $$\Q(\sqrt{6}, \sqrt{10})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database $4$ $$\Q(i, \sqrt{10})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(i, \sqrt{15})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database $6$ 6.0.472392000.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $8$ 8.0.3317760000.5 $$\Z/4\Z \times \Z/12\Z$$ Not in database $8$ 8.0.11664000000.11 $$\Z/2\Z \times \Z/8\Z$$ Not in database $12$ Deg 12 $$\Z/6\Z \times \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/24\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/24\Z$$ Not in database $18$ 18.6.316248778897380864000000000000000.3 $$\Z/2\Z \times \Z/18\Z$$ 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.