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Properties

Label 32340u2
Conductor $32340$
Discriminant $51744000$
j-invariant \( \frac{462893166690304}{4125} \)
CM no
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -37445, 2801457])
 
gp: E = ellinit([0, -1, 0, -37445, 2801457])
 
magma: E := EllipticCurve([0, -1, 0, -37445, 2801457]);
 

\(y^2=x^3-x^2-37445x+2801457\)  Toggle raw display

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \(\left(112, 1\right)\)  Toggle raw display
\(\hat{h}(P)\) ≈  $1.1095163726743367726340746118$

Integral points

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

\((112,\pm 1)\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 32340 \)  =  \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(51744000 \)  =  \(2^{8} \cdot 3 \cdot 5^{3} \cdot 7^{2} \cdot 11 \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{462893166690304}{4125} \)  =  \(2^{16} \cdot 3^{-1} \cdot 5^{-3} \cdot 7 \cdot 11^{-1} \cdot 17^{3} \cdot 59^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(1.0632870542693607098632166064\dots\)
Stable Faltings height: \(0.27687057572017828606750306819\dots\)

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1.1095163726743367726340746118\dots\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(1.3897940913519513706356636623\dots\)
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: \( 3 \)  = \( 1\cdot1\cdot3\cdot1\cdot1 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(1\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 32340.2.a.r

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 46656
\( \Gamma_0(N) \)-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);
 

\( L'(E,1) \) ≈ \( 4.6259978970031287661688700976581291334 \)

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\((j)_{-}\)
\(2\) \(1\) \(IV^{*}\) Additive -1 2 8 0
\(3\) \(1\) \(I_{1}\) Non-split multiplicative 1 1 1 1
\(5\) \(3\) \(I_{3}\) Split multiplicative -1 1 3 3
\(7\) \(1\) \(II\) Additive -1 2 2 0
\(11\) \(1\) \(I_{1}\) Split multiplicative -1 1 1 1

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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
\(3\) B

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 nonsplit split add split ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) - 3 2 - 2 1 1,1 1 1 1 1 1 1 1 1
$\mu$-invariant(s) - 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 \(d=\) 3.
Its isogeny class 32340u consists of 2 curves linked by isogenies of degree 3.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{21}) \) \(\Z/3\Z\) Not in database
$3$ 3.3.32340.1 \(\Z/2\Z\) Not in database
$6$ 6.6.172569474000.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$6$ 6.0.232484214817008.5 \(\Z/3\Z\) Not in database
$6$ 6.6.21963387600.1 \(\Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) Not in database
$12$ Deg 12 \(\Z/3\Z \times \Z/3\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/6\Z\) Not in database
$18$ 18.6.628595080445471730818700030794121501000000000000.4 \(\Z/9\Z\) Not in database
$18$ 18.0.213810149627339586224814152688703618643147712000000.1 \(\Z/6\Z\) 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.