Properties

Label 182070.bi8
Conductor 182070
Discriminant -4736305493311157226562500
j-invariant \( \frac{226523624554079}{269165039062500} \)
CM no
Rank 2
Torsion Structure \(\Z/{2}\Z\)

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Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, 3303216, 104681219188]) # or
 
sage: E = EllipticCurve("182070cj6")
 
gp: E = ellinit([1, -1, 0, 3303216, 104681219188]) \\ or
 
gp: E = ellinit("182070cj6")
 
magma: E := EllipticCurve([1, -1, 0, 3303216, 104681219188]); // or
 
magma: E := EllipticCurve("182070cj6");
 

\( y^2 + x y = x^{3} - x^{2} + 3303216 x + 104681219188 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(40677, -8238776\right) \)\( \left(302, -325276\right) \)
\(\hat{h}(P)\) ≈  2.0517192847361340.7210090561550054

Torsion generators

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

\( \left(-\frac{17917}{4}, \frac{17917}{8}\right) \)

Integral points

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

\( \left(-3073, 257474\right) \), \( \left(-3073, -254401\right) \), \( \left(-2758, 274484\right) \), \( \left(-2758, -271726\right) \), \( \left(-573, 320599\right) \), \( \left(-573, -320026\right) \), \( \left(302, 324974\right) \), \( \left(302, -325276\right) \), \( \left(3192, 382774\right) \), \( \left(3192, -385966\right) \), \( \left(8802, 898724\right) \), \( \left(8802, -907526\right) \), \( \left(11777, 1327124\right) \), \( \left(11777, -1338901\right) \), \( \left(40677, 8198099\right) \), \( \left(40677, -8238776\right) \), \( \left(56777, 13511099\right) \), \( \left(56777, -13567876\right) \), \( \left(2826552, 4750685974\right) \), \( \left(2826552, -4753512526\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 182070 \)  =  \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-4736305493311157226562500 \)  =  \(-1 \cdot 2^{2} \cdot 3^{8} \cdot 5^{16} \cdot 7^{2} \cdot 17^{6} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{226523624554079}{269165039062500} \)  =  \(2^{-2} \cdot 3^{-2} \cdot 5^{-16} \cdot 7^{-2} \cdot 47^{3} \cdot 1297^{3}\)
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: \(1.39300891207\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.0603384713115\)
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: \( 512 \)  = \( 2\cdot2\cdot2^{4}\cdot2\cdot2^{2} \)
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 182070.2.a.bi

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);
 

\( q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - q^{10} - 4q^{11} - 2q^{13} - q^{14} + q^{16} + 4q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 41943040
\( \Gamma_0(N) \)-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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 10.7586596195 \)

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\((j)_{-}\)
\(2\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(3\) \(2\) \( I_2^{*} \) Additive -1 2 8 2
\(5\) \(16\) \( I_{16} \) Split multiplicative -1 1 16 16
\(7\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2
\(17\) \(4\) \( I_0^{*} \) Additive 1 2 6 0

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 X207.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 8 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 14 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 3 \end{array}\right)$ and has index 48.

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

$p$-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.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 182070.bi consists of 8 curves linked by isogenies of degrees dividing 16.

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 \(\Q(\sqrt{-1}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
\(\Q(\sqrt{-51}) \) \(\Z/4\Z\) Not in database
\(\Q(\sqrt{51}) \) \(\Z/4\Z\) Not in database
4 \(\Q(i, \sqrt{51})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{-6}, \sqrt{34})\) \(\Z/8\Z\) Not in database
\(\Q(\sqrt{6}, \sqrt{-34})\) \(\Z/8\Z\) Not in database
\(\Q(\sqrt{14}, \sqrt{51})\) \(\Z/8\Z\) Not in database
\(\Q(\sqrt{-14}, \sqrt{51})\) \(\Z/8\Z\) 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.