Properties

Label 960.p4
Conductor $960$
Discriminant $8847360000$
j-invariant \( \frac{2656166199049}{33750} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3+x^2-18465x-971937\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3+x^2z-18465xz^2-971937z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-1495692x-704055024\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 0, -18465, -971937])
 
gp: E = ellinit([0, 1, 0, -18465, -971937])
 
magma: E := EllipticCurve([0, 1, 0, -18465, -971937]);
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 

Mordell-Weil group structure

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

Torsion generators

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

\( \left(-79, 0\right) \) Copy content Toggle raw display

Integral points

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

\( \left(-79, 0\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 960 \)  =  $2^{6} \cdot 3 \cdot 5$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $8847360000 $  =  $2^{19} \cdot 3^{3} \cdot 5^{4} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{2656166199049}{33750} \)  =  $2^{-1} \cdot 3^{-3} \cdot 5^{-4} \cdot 11^{3} \cdot 1259^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.0545806117222147987174678853\dots$
Stable Faltings height: $0.014859840882296834591619703113\dots$

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: $0.40955345859159473896425957380\dots$
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: $ 24 $  = $ 2\cdot3\cdot2^{2} $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $2$
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) $ ≈ $ 2.4573207515495684337855574428 $

Modular invariants

Modular form   960.2.a.p

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);
 

\( q + q^{3} + q^{5} + 4 q^{7} + q^{9} - 2 q^{13} + q^{15} + 6 q^{17} - 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 1536
$ \Gamma_0(N) $-optimal: no
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$(j)_{-}$
$2$ $2$ $I_{9}^{*}$ Additive -1 6 19 1
$3$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$5$ $4$ $I_{4}$ Split multiplicative -1 1 4 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
$2$ 2B 4.12.0.8
$3$ 3B 3.4.0.1

The image of the adelic Galois representation has level $120$, index $384$, and genus $5$.

$p$-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 $p\ge 5$ 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, 3, 4, 6 and 12.
Its isogeny class 960.p consists of 8 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{6}) \) \(\Z/2\Z \oplus \Z/2\Z\) 2.2.24.1-150.1-b10
$2$ \(\Q(\sqrt{-6}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{-1}) \) \(\Z/4\Z\) 2.0.4.1-57600.2-ba7
$2$ \(\Q(\sqrt{-2}) \) \(\Z/6\Z\) 2.0.8.1-450.2-a7
$4$ \(\Q(i, \sqrt{6})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$4$ 4.2.55296.1 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-2}, \sqrt{-3})\) \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$4$ \(\Q(\sqrt{-2}, \sqrt{3})\) \(\Z/12\Z\) Not in database
$4$ \(\Q(\zeta_{8})\) \(\Z/12\Z\) Not in database
$6$ 6.2.34560000.1 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$8$ 8.0.12230590464.4 \(\Z/4\Z \oplus \Z/4\Z\) Not in database
$8$ 8.0.7464960000.2 \(\Z/8\Z\) Not in database
$8$ 8.0.94371840000.17 \(\Z/8\Z\) Not in database
$8$ \(\Q(\zeta_{24})\) \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$8$ 8.0.3057647616.9 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$12$ 12.0.1194393600000000.1 \(\Z/6\Z \oplus \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$16$ 16.0.891610044825600000000.1 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ 16.0.149587343098087735296.14 \(\Z/4\Z \oplus \Z/12\Z\) Not in database
$16$ 16.0.891610044825600000000.8 \(\Z/24\Z\) Not in database
$16$ 16.0.8906044184985600000000.3 \(\Z/24\Z\) Not in database
$18$ 18.0.172713999781822935859200000000.1 \(\Z/18\Z\) 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.