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Properties

Label 179520.gd4
Conductor $179520$
Discriminant $3.776\times 10^{23}$
j-invariant \( \frac{1696892787277117093383481}{1440538624914939000} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -159034081, 771320361119])
 
gp: E = ellinit([0, 1, 0, -159034081, 771320361119])
 
magma: E := EllipticCurve([0, 1, 0, -159034081, 771320361119]);
 

Minimal equation

Minimal equation

Simplified equation

\(y^2=x^3+x^2-159034081x+771320361119\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3+x^2z-159034081xz^2+771320361119z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-12881760588x+562331188537488\) Copy content Toggle raw display (homogenize, minimize)

Mordell-Weil group structure

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

Torsion generators

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

\( \left(7441, 0\right) \) Copy content Toggle raw display

Integral points

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

\( \left(7441, 0\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 179520 \)  =  $2^{6} \cdot 3 \cdot 5 \cdot 11 \cdot 17$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $377628557289701769216000 $  =  $2^{21} \cdot 3^{3} \cdot 5^{3} \cdot 11^{12} \cdot 17 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{1696892787277117093383481}{1440538624914939000} \)  =  $2^{-3} \cdot 3^{-3} \cdot 5^{-3} \cdot 11^{-12} \cdot 17^{-1} \cdot 181^{3} \cdot 227^{3} \cdot 2903^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $3.4512020754179251972426445387\dots$
Stable Faltings height: $2.4114813045780072331167963565\dots$

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: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.094587733752222618555120887597\dots$
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: $ 144 $  = $ 2^{2}\cdot3\cdot1\cdot( 2^{2} \cdot 3 )\cdot1 $
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) $ ≈ $ 3.4051584150800142679843519535 $

Modular invariants

Modular form 179520.2.a.gd

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);
 

\( q + q^{3} - q^{5} + 4 q^{7} + q^{9} + q^{11} - 2 q^{13} - q^{15} - q^{17} - 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 42467328
$ \Gamma_0(N) $-optimal: no
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$(j)_{-}$
$2$ $4$ $I_{11}^{*}$ Additive -1 6 21 3
$3$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$5$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$11$ $12$ $I_{12}$ Split multiplicative -1 1 12 12
$17$ $1$ $I_{1}$ 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
$2$ 2B 4.6.0.1
$3$ 3B 3.4.0.1

$p$-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 $0$.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3, 4, 6 and 12.
Its isogeny class 179520.gd 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
$2$ \(\Q(\sqrt{510}) \) \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{3}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{170}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{-2}) \) \(\Z/6\Z\) Not in database
$4$ \(\Q(\sqrt{3}, \sqrt{170})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-2}, \sqrt{-255})\) \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$4$ \(\Q(\sqrt{-2}, \sqrt{3})\) \(\Z/12\Z\) Not in database
$4$ \(\Q(\sqrt{-2}, \sqrt{-85})\) \(\Z/12\Z\) Not in database
$6$ 6.2.1154594304.1 \(\Z/6\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/8\Z\) Not in database
$8$ Deg 8 \(\Z/8\Z\) Not in database
$8$ 8.0.277102632960000.157 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/3\Z \oplus \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/12\Z\) Not in database
$12$ Deg 12 \(\Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \oplus \Z/4\Z\) 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
$16$ 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
$18$ 18.0.147703630239956285761141491820151691313152000000000000.1 \(\Z/18\Z\) 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.