Properties

Label 4830bc2
Conductor 4830
Discriminant 5418996968417034240000
j-invariant \( \frac{9378698233516887309850369}{5418996968417034240000} \)
CM no
Rank 0
Torsion Structure \(\Z/{2}\Z \times \Z/{6}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -4393456, -140558080]); // or
magma: E := EllipticCurve("4830bc2");
sage: E = EllipticCurve([1, 0, 0, -4393456, -140558080]) # or
sage: E = EllipticCurve("4830bc2")
gp: E = ellinit([1, 0, 0, -4393456, -140558080]) \\ or
gp: E = ellinit("4830bc2")

\( y^2 + x y = x^{3} - 4393456 x - 140558080 \)

Mordell-Weil group structure

\(\Z/{2}\Z \times \Z/{6}\Z\)

Torsion generators

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

\( \left(-32, 16\right) \), \( \left(11168, 1153616\right) \)

Integral points

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

\( \left(-2080, 1040\right) \), \( \left(-1432, 57416\right) \), \( \left(-424, 40784\right) \), \( \left(-32, 16\right) \), \( \left(3104, 125456\right) \), \( \left(11168, 1153616\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 4830 \)  =  \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 23\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(5418996968417034240000 \)  =  \(2^{18} \cdot 3^{12} \cdot 5^{4} \cdot 7^{6} \cdot 23^{2} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{9378698233516887309850369}{5418996968417034240000} \)  =  \(2^{-18} \cdot 3^{-12} \cdot 5^{-4} \cdot 7^{-6} \cdot 23^{-2} \cdot 43^{3} \cdot 193^{3} \cdot 25411^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
Rank: \(0\)
magma: Regulator(E);
sage: E.regulator()
Regulator: \(1\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(0.113958070835\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
Tamagawa product: \( 5184 \)  = \( ( 2 \cdot 3^{2} )\cdot( 2^{2} \cdot 3 )\cdot2\cdot( 2 \cdot 3 )\cdot2 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
Torsion order: \(12\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 4830.2.a.be

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

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

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
331776 . This curve is not \( \Gamma_0(N) \)-optimal.

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar[2]/factorial(ar[1])

\( L(E,1) \) ≈ \( 4.10249055007 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(18\) \( I_{18} \) Split multiplicative -1 1 18 18
\(3\) \(12\) \( I_{12} \) Split multiplicative -1 1 12 12
\(5\) \(2\) \( I_{4} \) Non-split multiplicative 1 1 4 4
\(7\) \(6\) \( I_{6} \) Split multiplicative -1 1 6 6
\(23\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X8c.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 6 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 5 \end{array}\right)$ and has index 12.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime Image of Galois representation
\(2\) Cs
\(3\) B.1.1

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$ 2 3 5 7 23
Reduction type split split nonsplit split nonsplit
$\lambda$-invariant(s) 4 7 0 1 0
$\mu$-invariant(s) 0 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 4830.be consists of 8 curves linked by isogenies of degrees dividing 12.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
4 \(\Q(\sqrt{2}, \sqrt{-7})\) \(\Z/2\Z \times \Z/12\Z\) Not in database
\(\Q(\sqrt{7}, \sqrt{23})\) \(\Z/2\Z \times \Z/12\Z\) Not in database
\(\Q(\sqrt{-2}, \sqrt{-23})\) \(\Z/2\Z \times \Z/12\Z\) Not in database
6 6.0.4722316875.1 \(\Z/6\Z \times \Z/6\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.