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Properties

Label 4830ba1
Conductor $4830$
Discriminant $-757170892800$
j-invariant \( -\frac{164287467238609}{757170892800} \)
CM no
Rank $1$
Torsion structure \(\Z/{4}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -1141, 44321])
 
gp: E = ellinit([1, 0, 0, -1141, 44321])
 
magma: E := EllipticCurve([1, 0, 0, -1141, 44321]);
 

Minimal equation

Minimal equation

Simplified equation

\(y^2+xy=x^3-1141x+44321\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-1141xz^2+44321z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-1478763x+2072276838\) Copy content Toggle raw display (homogenize, minimize)

Mordell-Weil group structure

\(\Z \oplus \Z/{4}\Z\)

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(8, 185\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.31740833233875293497377028291$

Torsion generators

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

\( \left(26, 167\right) \) Copy content Toggle raw display

Integral points

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

\( \left(-46, 23\right) \), \( \left(-34, 227\right) \), \( \left(-34, -193\right) \), \( \left(-14, 247\right) \), \( \left(-14, -233\right) \), \( \left(-10, 239\right) \), \( \left(-10, -229\right) \), \( \left(8, 185\right) \), \( \left(8, -193\right) \), \( \left(26, 167\right) \), \( \left(26, -193\right) \), \( \left(50, 311\right) \), \( \left(50, -361\right) \), \( \left(98, 887\right) \), \( \left(98, -985\right) \), \( \left(116, 1157\right) \), \( \left(116, -1273\right) \), \( \left(386, 7367\right) \), \( \left(386, -7753\right) \), \( \left(554, 12743\right) \), \( \left(554, -13297\right) \), \( \left(14066, 1661207\right) \), \( \left(14066, -1675273\right) \) Copy content 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: $-757170892800 $  =  $-1 \cdot 2^{12} \cdot 3^{8} \cdot 5^{2} \cdot 7^{2} \cdot 23 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{164287467238609}{757170892800} \)  =  $-1 \cdot 2^{-12} \cdot 3^{-8} \cdot 5^{-2} \cdot 7^{-2} \cdot 11^{3} \cdot 13^{3} \cdot 23^{-1} \cdot 383^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.96413422269230153213547638457\dots$
Stable Faltings height: $0.96413422269230153213547638457\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.31740833233875293497377028291\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.78114409035795583736095163298\dots$
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: $ 384 $  = $ ( 2^{2} \cdot 3 )\cdot2^{3}\cdot2\cdot2\cdot1 $
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) $ ≈ $ 5.9505994328829815643018358023 $

Modular invariants

Modular form   4830.2.a.ba

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 9216
$ \Gamma_0(N) $-optimal: yes
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$(j)_{-}$
$2$ $12$ $I_{12}$ Split multiplicative -1 1 12 12
$3$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$7$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$23$ $1$ $I_{1}$ 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 for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 8.24.0.50

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type split split nonsplit nonsplit ss ord ord ss nonsplit ord ord ord ord ss ord
$\lambda$-invariant(s) 4 2 1 1 1,1 1 3 1,1 1 3 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 0

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 4830ba consists of 4 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-23}) \) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$4$ 4.2.36800.3 \(\Z/8\Z\) Not in database
$8$ 8.0.56869467118240000.26 \(\Z/4\Z \oplus \Z/4\Z\) Not in database
$8$ 8.0.716392960000.7 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$8$ 8.0.1455858358226944.43 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$8$ 8.2.74390473561516875.1 \(\Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/12\Z\) 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.