Properties

Label 106722.bu1
Conductor 106722
Discriminant 38828549179771509312
j-invariant \( \frac{57736239625}{255552} \)
CM no
Rank 2
Torsion Structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -4296672, 3415983808]) # or
 
sage: E = EllipticCurve("106722cu3")
 
gp: E = ellinit([1, -1, 0, -4296672, 3415983808]) \\ or
 
gp: E = ellinit("106722cu3")
 
magma: E := EllipticCurve([1, -1, 0, -4296672, 3415983808]); // or
 
magma: E := EllipticCurve("106722cu3");
 

\( y^2 + x y = x^{3} - x^{2} - 4296672 x + 3415983808 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

sage: E.gens()
 
magma: Generators(E);
 

\(P\) =  \( \left(597, 32311\right) \)\( \left(-1768, 74936\right) \)
\(\hat{h}(P)\) ≈  1.02528123786322123.247424834645178

Torsion generators

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

\( \left(1136, -568\right) \)

Integral points

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

\( \left(-2392, 2960\right) \), \( \left(-2392, -568\right) \), \( \left(-1768, 74936\right) \), \( \left(-1768, -73168\right) \), \( \left(597, 32311\right) \), \( \left(597, -32908\right) \), \( \left(1112, 2936\right) \), \( \left(1112, -4048\right) \), \( \left(1136, -568\right) \), \( \left(1257, -568\right) \), \( \left(1257, -689\right) \), \( \left(1283, 3107\right) \), \( \left(1283, -4390\right) \), \( \left(1928, 46952\right) \), \( \left(1928, -48880\right) \), \( \left(18923, 2578547\right) \), \( \left(18923, -2597470\right) \), \( \left(13316337, 48586726352\right) \), \( \left(13316337, -48600042689\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 106722 \)  =  \(2 \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(38828549179771509312 \)  =  \(2^{6} \cdot 3^{7} \cdot 7^{6} \cdot 11^{9} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{57736239625}{255552} \)  =  \(2^{-6} \cdot 3^{-1} \cdot 5^{3} \cdot 11^{-3} \cdot 773^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(2\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(2.6810545638\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.205744150159\)
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: \( 64 \)  = \( 2\cdot2^{2}\cdot2\cdot2^{2} \)
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: \(1\) (rounded)

Modular invariants

Modular form 106722.2.a.bu

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);
 

\( q - q^{2} + q^{4} - q^{8} - 4q^{13} + q^{16} + 6q^{17} - 4q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 4147200
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 8.82578068414 \)

Local data

This elliptic curve is not semistable.

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\((j)_{-}\)
\(2\) \(2\) \( I_{6} \) Non-split multiplicative 1 1 6 6
\(3\) \(4\) \( I_1^{*} \) Additive -1 2 7 1
\(7\) \(2\) \( I_0^{*} \) Additive -1 2 6 0
\(11\) \(4\) \( I_3^{*} \) Additive -1 2 9 3

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X15.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 2 & 1 \end{array}\right)$ and has index 6.

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 mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime Image of Galois representation
\(2\) B
\(3\) B

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
 

\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit add ss add add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 8 - 4,2 - - 2 2 2 2 4 2 2 2 2 2
$\mu$-invariant(s) 0 - 0,0 - - 0 0 0 0 0 0 0 0 0 0

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 and 6.
Its isogeny class 106722.bu consists of 4 curves linked by isogenies of degrees dividing 6.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 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 \(\Q(\sqrt{33}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
\(\Q(\sqrt{77}) \) \(\Z/6\Z\) Not in database
4 \(\Q(\sqrt{21}, \sqrt{33})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
4.0.103488.1 \(\Z/4\Z\) Not in database
6 6.0.80873451351.4 \(\Z/6\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.