Properties

Label 4830y2
Conductor $4830$
Discriminant $2.745\times 10^{20}$
j-invariant \( \frac{1567558142704512417614401}{274462175610000000000} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -2420100, 1209181317])
 
gp: E = ellinit([1, 1, 1, -2420100, 1209181317])
 
magma: E := EllipticCurve([1, 1, 1, -2420100, 1209181317]);
 

\(y^2+xy+y=x^3+x^2-2420100x+1209181317\)  Toggle raw display

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(537, 7781\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.15256791665967643513813438922$

Torsion generators

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

\( \left(-1763, 881\right) \), \( \left(1181, -591\right) \)  Toggle raw display

Integral points

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

\( \left(-1763, 881\right) \), \( \left(-1563, 35081\right) \), \( \left(-1563, -33519\right) \), \( \left(-1073, 51251\right) \), \( \left(-1073, -50179\right) \), \( \left(-513, 48381\right) \), \( \left(-513, -47869\right) \), \( \left(-263, 42881\right) \), \( \left(-263, -42619\right) \), \( \left(285, 23153\right) \), \( \left(285, -23439\right) \), \( \left(397, 17441\right) \), \( \left(397, -17839\right) \), \( \left(537, 7781\right) \), \( \left(537, -8319\right) \), \( \left(1181, -591\right) \), \( \left(1237, 9881\right) \), \( \left(1237, -11119\right) \), \( \left(1867, 55661\right) \), \( \left(1867, -57529\right) \), \( \left(2837, 129681\right) \), \( \left(2837, -132519\right) \), \( \left(3757, 210641\right) \), \( \left(3757, -214399\right) \), \( \left(8237, 730881\right) \), \( \left(8237, -739119\right) \), \( \left(32737, 5900381\right) \), \( \left(32737, -5933119\right) \), \( \left(132907, 48383591\right) \), \( \left(132907, -48516499\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 4830 \)  =  $2 \cdot 3 \cdot 5 \cdot 7 \cdot 23$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $274462175610000000000 $  =  $2^{10} \cdot 3^{2} \cdot 5^{10} \cdot 7^{8} \cdot 23^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{1567558142704512417614401}{274462175610000000000} \)  =  $2^{-10} \cdot 3^{-2} \cdot 5^{-10} \cdot 7^{-8} \cdot 23^{-2} \cdot 43^{3} \cdot 61^{3} \cdot 67^{3} \cdot 661^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.6416051430208161576935555178\dots$
Stable Faltings height: $2.6416051430208161576935555178\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.15256791665967643513813438922\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.16575857733038252567834586680\dots$
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: $ 3200 $  = $ ( 2 \cdot 5 )\cdot2\cdot( 2 \cdot 5 )\cdot2^{3}\cdot2 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $4$
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) $ ≈ $ 5.0578881623536665624874473577151360968 $

Modular invariants

Modular form   4830.2.a.z

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);
 

\( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - q^{12} - 6q^{13} + q^{14} - q^{15} + q^{16} - 6q^{17} + q^{18} + 4q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 307200
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is 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$(j)_{-}$
$2$ $10$ $I_{10}$ Split multiplicative -1 1 10 10
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$5$ $10$ $I_{10}$ Split multiplicative -1 1 10 10
$7$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$23$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

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
$2$ 2Cs 4.12.0.1

$p$-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 split nonsplit split split ss ordinary ordinary ordinary split ordinary ss ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 4 1 2 2 1,1 1 1 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 0

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 4830y consists of 2 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{46}) \) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(i, \sqrt{15})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-15}, \sqrt{-46})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.928445276160000.208 \(\Z/4\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/6\Z\) 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 \(\Z/2\Z \times \Z/12\Z\) 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.