Properties

Label 187425bh4
Conductor 187425
Discriminant 1794390320858029453125
j-invariant \( \frac{17032120495489}{1339001685} \)
CM no
Rank 2
Torsion Structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -5909630, -5138762128]) # or
 
sage: E = EllipticCurve("187425bh4")
 
gp: E = ellinit([1, -1, 1, -5909630, -5138762128]) \\ or
 
gp: E = ellinit("187425bh4")
 
magma: E := EllipticCurve([1, -1, 1, -5909630, -5138762128]); // or
 
magma: E := EllipticCurve("187425bh4");
 

\( y^2 + x y + y = x^{3} - x^{2} - 5909630 x - 5138762128 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(4734, -272480\right) \)\( \left(\frac{66111}{4}, -\frac{16869565}{8}\right) \)
\(\hat{h}(P)\) ≈  2.11892371134124426.274304388660094

Torsion generators

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

\( \left(-\frac{6789}{4}, \frac{6785}{8}\right) \)

Integral points

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

\( \left(-1566, 17320\right) \), \( \left(-1566, -15755\right) \), \( \left(-1272, 18496\right) \), \( \left(-1272, -17225\right) \), \( \left(-1097, 5350\right) \), \( \left(-1097, -4254\right) \), \( \left(2859, 35020\right) \), \( \left(2859, -37880\right) \), \( \left(3415, 118698\right) \), \( \left(3415, -122114\right) \), \( \left(4734, 267745\right) \), \( \left(4734, -272480\right) \), \( \left(19140, 2615923\right) \), \( \left(19140, -2635064\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 187425 \)  =  \(3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 17\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(1794390320858029453125 \)  =  \(3^{14} \cdot 5^{7} \cdot 7^{10} \cdot 17 \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{17032120495489}{1339001685} \)  =  \(3^{-8} \cdot 5^{-1} \cdot 7^{-4} \cdot 11^{3} \cdot 17^{-1} \cdot 2339^{3}\)
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: \(5.31001202434\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.0973124407198\)
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: \( 64 \)  = \( 2^{2}\cdot2^{2}\cdot2^{2}\cdot1 \)
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: \(1\) (rounded)

Modular invariants

Modular form 187425.2.a.bl

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);
 

\( q - q^{2} - q^{4} + 3q^{8} - 6q^{13} - q^{16} + q^{17} - 4q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 9437184
\( \Gamma_0(N) \)-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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 8.26768368544 \)

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\((j)_{-}\)
\(3\) \(4\) \( I_8^{*} \) Additive -1 2 14 8
\(5\) \(4\) \( I_1^{*} \) Additive 1 2 7 1
\(7\) \(4\) \( I_4^{*} \) Additive -1 2 10 4
\(17\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1

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 X13.

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

$p$-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 and 4.
Its isogeny class 187425bh consists of 4 curves linked by isogenies of degrees dividing 4.

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 \(\Q(\sqrt{-21}) \) \(\Z/4\Z\) Not in database
\(\Q(\sqrt{85}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
\(\Q(\sqrt{-1785}) \) \(\Z/4\Z\) Not in database
4 \(\Q(\sqrt{-21}, \sqrt{85})\) \(\Z/2\Z \times \Z/4\Z\) 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.