Properties

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

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3+x^2-15x+18\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3+x^2z-15xz^2+18z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-1242x+16821\) Copy content Toggle raw display (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);
 

\( \left(3, 3\right) \) Copy content Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(2, 0\right) \), \((3,\pm 3)\) Copy content Toggle raw display

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

Modular form   120.2.a.b

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);
 

\( q + q^{3} + q^{5} + q^{9} - 4 q^{11} + 6 q^{13} + q^{15} - 6 q^{17} - 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

For more coefficients, see the Downloads section to the right.

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

$p$ 2 3 5
Reduction type add split split
$\lambda$-invariant(s) - 3 1
$\mu$-invariant(s) - 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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{5}) \) \(\Z/2\Z \oplus \Z/4\Z\) 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.