# Properties

 Label 369600.hb1 Conductor $369600$ Discriminant $-2.376\times 10^{18}$ j-invariant $$-\frac{4890195460096}{9282994875}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3-x^2-224533x-84648563$$ y^2=x^3-x^2-224533x-84648563 (homogenize, simplify) $$y^2z=x^3-x^2z-224533xz^2-84648563z^3$$ y^2z=x^3-x^2z-224533xz^2-84648563z^3 (dehomogenize, simplify) $$y^2=x^3-18187200x-61763364000$$ y^2=x^3-18187200x-61763364000 (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 0, -224533, -84648563])

gp: E = ellinit([0, -1, 0, -224533, -84648563])

magma: E := EllipticCurve([0, -1, 0, -224533, -84648563]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(3732, 225925\right)$$ (3732, 225925) $\hat{h}(P)$ ≈ $5.3997675226874591185859133421$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(3732,\pm 225925)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$369600$$ = $2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-2376446688000000000$ = $-1 \cdot 2^{14} \cdot 3^{9} \cdot 5^{9} \cdot 7^{3} \cdot 11$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{4890195460096}{9282994875}$$ = $-1 \cdot 2^{16} \cdot 3^{-9} \cdot 5^{-3} \cdot 7^{-3} \cdot 11^{-1} \cdot 421^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.2158003282811961401911936476\dots$ Stable Faltings height: $0.60240966141087642523737650595\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $5.3997675226874591185859133421\dots$ sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $0.10323993917161500320661702444\dots$ 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: $6$  = $1\cdot1\cdot2\cdot3\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $1$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) 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); Special value: $L'(E,1)$ ≈ $3.3448300234986930971327893355$

## Modular invariants

Modular form 369600.2.a.hb

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);

$$q - q^{3} + q^{7} + q^{9} - q^{11} - 4 q^{13} - 3 q^{17} - 7 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 5971968 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 5 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$(j)_{-}$
$2$ $1$ $II^{*}$ Additive -1 6 14 0
$3$ $1$ $I_{9}$ Non-split multiplicative 1 1 9 9
$5$ $2$ $I_{3}^{*}$ Additive 1 2 9 3
$7$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$11$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

## Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$3$ 3B 3.4.0.1

The image of the adelic Galois representation has level $9240$, index $16$, and genus $0$.

## $p$-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.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 369600.hb 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-10})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.4620.1 $$\Z/2\Z$$ Not in database $6$ 6.0.13660416000.5 $$\Z/6\Z$$ Not in database $6$ 6.0.24652782000.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $6$ 6.2.101198592000.14 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \oplus \Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $18$ 18.0.21770278665850141802496000000000000000.1 $$\Z/9\Z$$ Not in database $18$ 18.2.14753566573346798204580582653952000000000.1 $$\Z/6\Z$$ 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.