Properties

Label 152880hv3
Conductor $152880$
Discriminant $46984304640$
j-invariant \( \frac{31103978031362}{195} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -407696, 100332576])
 
gp: E = ellinit([0, -1, 0, -407696, 100332576])
 
magma: E := EllipticCurve([0, -1, 0, -407696, 100332576]);
 

\(y^2=x^3-x^2-407696x+100332576\)  Toggle raw display

Mordell-Weil group structure

$\Z^2 \times \Z/{2}\Z$

Infinite order Mordell-Weil generators and heights

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

$P$ =  \(\left(370, 34\right)\)  Toggle raw display\(\left(373, 134\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $2.0467308420561336183474769963$$3.2939528777389024539829313864$

Torsion generators

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

\( \left(369, 0\right) \)  Toggle raw display

Integral points

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

\( \left(369, 0\right) \), \((370,\pm 34)\), \((373,\pm 134)\), \((418,\pm 1666)\), \((810,\pm 17346)\), \((2394,\pm 113310)\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 152880 \)  =  $2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $46984304640 $  =  $2^{11} \cdot 3 \cdot 5 \cdot 7^{6} \cdot 13 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{31103978031362}{195} \)  =  $2 \cdot 3^{-1} \cdot 5^{-1} \cdot 13^{-1} \cdot 109^{3} \cdot 229^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.6533339303822174820039472315\dots$
Stable Faltings height: $0.044993940341277629152141415108\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $2$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $6.0594564130815256274091373801\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.77567006711255727550571373955\dots$
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: $ 8 $  = $ 2^{2}\cdot1\cdot1\cdot2\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)
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^{(2)}(E,1)/2! $ ≈ $ 9.4002779252011251295534877355583262145 $

Modular invariants

Modular form 152880.2.a.e

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);
 

\( q - q^{3} - q^{5} + q^{9} - 4q^{11} - q^{13} + q^{15} - 6q^{17} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 884736
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

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$(j)_{-}$
$2$ $4$ $I_3^{*}$ Additive 1 4 11 0
$3$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$5$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$7$ $2$ $I_0^{*}$ Additive -1 2 6 0
$13$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

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 $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 4.6.0.1

$p$-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.

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 152880hv consists of 3 curves linked by isogenies of degrees dividing 4.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{390}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{70}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{273}) \) \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{70}, \sqrt{273})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/8\Z\) Not in database
$8$ Deg 8 \(\Z/8\Z\) Not in database
$8$ Deg 8 \(\Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \times \Z/4\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/12\Z\) 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.