Properties

Label 40432.b2
Conductor 40432
Discriminant -353605910536388608
j-invariant \( -\frac{548347731625}{1835008} \)
CM no
Rank 1
Torsion Structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 0, -984928, -377645964]); // or
 
magma: E := EllipticCurve("40432r5");
 
sage: E = EllipticCurve([0, 1, 0, -984928, -377645964]) # or
 
sage: E = EllipticCurve("40432r5")
 
gp: E = ellinit([0, 1, 0, -984928, -377645964]) \\ or
 
gp: E = ellinit("40432r5")
 

\( y^2 = x^{3} + x^{2} - 984928 x - 377645964 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(\frac{16210593284746}{27405225}, -\frac{65267671648151711744}{143466352875}\right) \)
\(\hat{h}(P)\) ≈  25.912198608

Torsion generators

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

\( \left(1146, 0\right) \)

Integral points

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

\( \left(1146, 0\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 40432 \)  =  \(2^{4} \cdot 7 \cdot 19^{2}\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(-353605910536388608 \)  =  \(-1 \cdot 2^{30} \cdot 7 \cdot 19^{6} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( -\frac{548347731625}{1835008} \)  =  \(-1 \cdot 2^{-18} \cdot 5^{3} \cdot 7^{-1} \cdot 1637^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(1\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(25.912198608\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.0757585031499\)
magma: TamagawaNumbers(E);
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Tamagawa product: \( 8 \)  = \( 2^{2}\cdot1\cdot2 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(2\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 40432.2.a.b

magma: ModularForm(E);
 
sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 

\( q - 2q^{3} - q^{7} + q^{9} + 4q^{13} + 6q^{17} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 513216
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 

\( L'(E,1) \) ≈ \( 3.92613875973 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(4\) \( I_22^{*} \) Additive -1 4 30 18
\(7\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(19\) \(2\) \( I_0^{*} \) 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 X16.

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

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 
sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 

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 add ordinary ss nonsplit ss ordinary ordinary add ss ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) - 3 1,1 1 3,1 1 1 - 1,1 1 1 1 1 1 1
$\mu$-invariant(s) - 2 0,0 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, 6, 9 and 18.
Its isogeny class 40432.b consists of 6 curves linked by isogenies of degrees dividing 18.

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 \(\Q(\sqrt{-57}) \) \(\Z/6\Z\) Not in database
\(\Q(\sqrt{-7}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
4 4.2.646912.1 \(\Z/4\Z\) Not in database
\(\Q(\sqrt{-7}, \sqrt{-57})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
6 6.0.20745515423808.8 \(\Z/18\Z\) Not in database
6.2.768352423104.2 \(\Z/6\Z\) 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.