# Properties

 Label 120.b5 Conductor $120$ Discriminant $720$ j-invariant $$\frac{24918016}{45}$$ CM no Rank $0$ Torsion structure $$\Z/{4}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3+x^2-15x+18$$ y^2=x^3+x^2-15x+18 (homogenize, simplify) $$y^2z=x^3+x^2z-15xz^2+18z^3$$ y^2z=x^3+x^2z-15xz^2+18z^3 (dehomogenize, simplify) $$y^2=x^3-1242x+16821$$ y^2=x^3-1242x+16821 (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 0, -15, 18])

gp: E = ellinit([0, 1, 0, -15, 18])

magma: E := EllipticCurve([0, 1, 0, -15, 18]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z/{4}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(3, 3\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(2, 0\right)$$, $$(3,\pm 3)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$120$$ = $2^{3} \cdot 3 \cdot 5$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $720$ = $2^{4} \cdot 3^{2} \cdot 5$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{24918016}{45}$$ = $2^{11} \cdot 3^{-2} \cdot 5^{-1} \cdot 23^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.55972190577000337273713211445\dots$ Stable Faltings height: $-0.79077096595665180920954282160\dots$

## 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: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $5.0779771118533319593856257082\dots$ 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: $4$  = $2\cdot2\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $4$ 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: $L(E,1)$ ≈ $1.2694942779633329898464064271$

## 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);

$$q + q^{3} + q^{5} + q^{9} - 4 q^{11} + 6 q^{13} + q^{15} - 6 q^{17} - 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 8 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## 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$(j)_{-}$
$2$ $2$ $III$ Additive -1 3 4 0
$3$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$5$ $1$ $I_{1}$ 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
$2$ 2B 16.48.0.30
sage: gens = [[5, 4, 236, 237], [1, 16, 180, 181], [173, 16, 64, 165], [112, 5, 195, 226], [225, 16, 224, 17], [1, 0, 16, 1], [15, 2, 142, 227], [1, 16, 0, 1], [45, 16, 74, 59]]

sage: GL(2,Integers(240)).subgroup(gens)

magma: Gens := [[5, 4, 236, 237], [1, 16, 180, 181], [173, 16, 64, 165], [112, 5, 195, 226], [225, 16, 224, 17], [1, 0, 16, 1], [15, 2, 142, 227], [1, 16, 0, 1], [45, 16, 74, 59]];

magma: sub<GL(2,Integers(240))|Gens>;

The image of the adelic Galois representation has level $240$, index $192$, genus $1$, and generators

$\left(\begin{array}{rr} 5 & 4 \\ 236 & 237 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 180 & 181 \end{array}\right),\left(\begin{array}{rr} 173 & 16 \\ 64 & 165 \end{array}\right),\left(\begin{array}{rr} 112 & 5 \\ 195 & 226 \end{array}\right),\left(\begin{array}{rr} 225 & 16 \\ 224 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 142 & 227 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 45 & 16 \\ 74 & 59 \end{array}\right)$

## $p$-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 split - 3 1 - 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ 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, 4 and 8.
Its isogeny class 120.b consists of 6 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{5})$$ $$\Z/2\Z \oplus \Z/4\Z$$ 2.2.5.1-2880.1-d6 $2$ $$\Q(\sqrt{3})$$ $$\Z/8\Z$$ 2.2.12.1-600.1-c5 $2$ $$\Q(\sqrt{15})$$ $$\Z/8\Z$$ 2.2.60.1-120.1-l5 $4$ $$\Q(\sqrt{3}, \sqrt{5})$$ $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.5184000000.15 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.64000000.3 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.19110297600.4 $$\Z/16\Z$$ Not in database $8$ 8.8.11943936000000.2 $$\Z/16\Z$$ Not in database $8$ 8.2.113374080000.2 $$\Z/12\Z$$ Not in database $16$ 16.0.26873856000000000000.6 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ 16.0.2123366400000000000000.1 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $16$ deg 16 $$\Z/24\Z$$ Not in database $16$ deg 16 $$\Z/24\Z$$ 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.