Label 72a4
Conductor $72$
Discriminant $60466176$
j-invariant \( \frac{1556068}{81} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z \oplus \Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-219x+1190\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-219xz^2+1190z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-219x+1190\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([0, 0, 0, -219, 1190])
gp: E = ellinit([0, 0, 0, -219, 1190])
magma: E := EllipticCurve([0, 0, 0, -219, 1190]);
sage: E.short_weierstrass_model()
magma: WeierstrassModel(E);

Mordell-Weil group structure

\(\Z/{2}\Z \oplus \Z/{2}\Z\)

Torsion generators

sage: E.torsion_subgroup().gens()
gp: elltors(E)
magma: TorsionSubgroup(E);

\( \left(7, 0\right) \), \( \left(10, 0\right) \) Copy content Toggle raw display

Integral points

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

\( \left(-17, 0\right) \), \( \left(7, 0\right) \), \( \left(10, 0\right) \) Copy content Toggle raw display


sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
Conductor: \( 72 \)  =  $2^{3} \cdot 3^{2}$
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
Discriminant: $60466176 $  =  $2^{10} \cdot 3^{10} $
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
j-invariant: \( \frac{1556068}{81} \)  =  $2^{2} \cdot 3^{-4} \cdot 73^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.25052744853417022774722800236\dots$
Stable Faltings height: $-0.87640134626650570913142138398\dots$

BSD invariants

sage: E.rank()
magma: Rank(E);
Analytic rank: $0$
sage: E.regulator()
magma: Regulator(E);
Regulator: $1$
sage: E.period_lattice().omega()
magma: RealPeriod(E);
Real period: $1.9465368422751189394858646060\dots$
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: $ 8 $  = $ 2\cdot2^{2} $
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) $ ≈ $ 0.97326842113755946974293230302 $

Modular invariants

Modular form   72.2.a.a

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);

\( q + 2 q^{5} - 4 q^{11} - 2 q^{13} - 2 q^{17} - 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
magma: ModularDegree(E);
Modular degree: 16
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 2 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 10 0
$3$ $4$ $I_{4}^{*}$ Additive -1 2 10 4

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
$2$ 2Cs

$p$-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
Reduction type add add
$\lambda$-invariant(s) - -
$\mu$-invariant(s) - -

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.


This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 72a 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{3}) \) \(\Z/2\Z \oplus \Z/8\Z\)
$4$ \(\Q(\sqrt{-2}, \sqrt{-3})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{2}, \sqrt{-3})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ \(\Q(\zeta_{24})\) \(\Z/4\Z \oplus \Z/8\Z\) Not in database
$8$ 8.4.764411904.2 \(\Z/2\Z \oplus \Z/16\Z\) Not in database
$8$ 8.2.181398528.2 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$16$ 16.0.1846757322198614016.5 \(\Z/8\Z \oplus \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/24\Z\) 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.