Properties

Label 342c3
Conductor $342$
Discriminant $60002532$
j-invariant \( \frac{8671983378625}{82308} \)
CM no
Rank $1$
Torsion structure \(\Z/{6}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -3852, 92988])
 
gp: E = ellinit([1, -1, 0, -3852, 92988])
 
magma: E := EllipticCurve([1, -1, 0, -3852, 92988]);
 

\(y^2+xy=x^3-x^2-3852x+92988\)  Toggle raw display

Mordell-Weil group structure

$\Z\times \Z/{6}\Z$

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(42, 42\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $1.0648513558535038321810741577$

Torsion generators

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

\( \left(18, 162\right) \)  Toggle raw display

Integral points

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

\( \left(-62, 332\right) \), \( \left(-62, -270\right) \), \( \left(-39, 447\right) \), \( \left(-39, -408\right) \), \( \left(18, 162\right) \), \( \left(18, -180\right) \), \( \left(33, 15\right) \), \( \left(33, -48\right) \), \( \left(36, -18\right) \), \( \left(37, -9\right) \), \( \left(37, -28\right) \), \( \left(42, 42\right) \), \( \left(42, -84\right) \), \( \left(132, 1302\right) \), \( \left(132, -1434\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 342 \)  =  $2 \cdot 3^{2} \cdot 19$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $60002532 $  =  $2^{2} \cdot 3^{7} \cdot 19^{3} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{8671983378625}{82308} \)  =  $2^{-2} \cdot 3^{-1} \cdot 5^{3} \cdot 7^{3} \cdot 19^{-3} \cdot 587^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.65480819028415759682295018776\dots$
Stable Faltings height: $0.10550204595010275112532756930\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1.0648513558535038321810741577\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $1.7816724979966209210767679034\dots$
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: $ 24 $  = $ 2\cdot2^{2}\cdot3 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $6$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
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);
 
Special value: $ L'(E,1) $ ≈ $ 1.2648109167857339186411249913487300188 $

Modular invariants

Modular form   342.2.a.c

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);
 

\( q - q^{2} + q^{4} - 4q^{7} - q^{8} - 4q^{13} + 4q^{14} + q^{16} - 6q^{17} + q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 288
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 3 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$(j)_{-}$
$2$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$3$ $4$ $I_1^{*}$ Additive -1 2 7 1
$19$ $3$ $I_{3}$ Split multiplicative -1 1 3 3

Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 8.6.0.4
$3$ 3B.1.1 3.8.0.1

$p$-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 nonsplit add ss ordinary ss ordinary ordinary split ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 1 - 1,1 1 1,3 3 1 2 1 1 1 1 1 1 1
$\mu$-invariant(s) 0 - 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 and 6.
Its isogeny class 342c consists of 4 curves linked by isogenies of degrees dividing 6.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{57}) \) \(\Z/2\Z \times \Z/6\Z\) 2.2.57.1-228.1-s7
$4$ 4.0.3648.1 \(\Z/12\Z\) Not in database
$6$ 6.0.2834352.1 \(\Z/3\Z \times \Z/6\Z\) Not in database
$8$ 8.4.35119561982976.11 \(\Z/2\Z \times \Z/12\Z\) Not in database
$8$ 8.0.43237380096.1 \(\Z/2\Z \times \Z/12\Z\) Not in database
$9$ 9.3.10800911539233072.2 \(\Z/18\Z\) Not in database
$12$ Deg 12 \(\Z/6\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$18$ 18.6.2400506442741963672809014465691175168.1 \(\Z/2\Z \times \Z/18\Z\) 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.