Properties

Label 30a5
Conductor $30$
Discriminant $33750$
j-invariant \( \frac{2656166199049}{33750} \)
CM no
Rank $0$
Torsion structure \(\Z/{6}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -289, 1862]) # or
 
sage: E = EllipticCurve("30.a4")
 
gp: E = ellinit([1, 0, 1, -289, 1862]) \\ or
 
gp: E = ellinit("30.a4")
 
magma: E := EllipticCurve([1, 0, 1, -289, 1862]); // or
 
magma: E := EllipticCurve("30.a4");
 

\(y^2+xy+y=x^3-289x+1862\)

Mordell-Weil group structure

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

Torsion generators

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

\( \left(16, 29\right) \)

Integral points

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

\( \left(10, -4\right) \), \( \left(10, -7\right) \), \( \left(16, 29\right) \), \( \left(16, -46\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 30 \)  =  \(2 \cdot 3 \cdot 5\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(33750 \)  =  \(2 \cdot 3^{3} \cdot 5^{4} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{2656166199049}{33750} \)  =  \(2^{-1} \cdot 3^{-3} \cdot 5^{-4} \cdot 11^{3} \cdot 1259^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(0\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(3.3519482592414964494482281339\)
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: \( 6 \)  = \( 1\cdot3\cdot2 \)
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)

Modular invariants

Modular form   30.2.a.a

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 8
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

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);
 

\( L(E,1) \) ≈ \( 0.55865804320691607490803802232401598477 \)

Local data

This elliptic curve is 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\((j)_{-}\)
\(2\) \(1\) \(I_{1}\) Non-split multiplicative 1 1 1 1
\(3\) \(3\) \(I_{3}\) Split multiplicative -1 1 3 3
\(5\) \(2\) \(I_{4}\) Non-split multiplicative 1 1 4 4

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 X13e.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 5 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 3 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$ and has index 12.

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

prime Image of Galois representation
\(2\) B
\(3\) 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
Reduction type nonsplit split nonsplit
$\lambda$-invariant(s) 0 1 0
$\mu$-invariant(s) 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, 4, 6 and 12.
Its isogeny class 30a consists of 6 curves linked by isogenies of degrees dividing 12.

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
$2$ \(\Q(\sqrt{6}) \) \(\Z/2\Z \times \Z/6\Z\) 2.2.24.1-150.1-e10
$2$ \(\Q(\sqrt{2}) \) \(\Z/12\Z\) 2.2.8.1-450.1-a7
$2$ \(\Q(\sqrt{3}) \) \(\Z/12\Z\) 2.2.12.1-150.1-a7
$4$ \(\Q(\sqrt{2}, \sqrt{3})\) \(\Z/2\Z \times \Z/12\Z\) Not in database
$6$ 6.0.270000.1 \(\Z/3\Z \times \Z/6\Z\) Not in database
$8$ 8.0.3057647616.9 \(\Z/2\Z \times \Z/12\Z\) Not in database
$8$ 8.8.23592960000.1 \(\Z/24\Z\) Not in database
$8$ 8.0.1866240000.6 \(\Z/24\Z\) Not in database
$9$ 9.3.143489070000.1 \(\Z/18\Z\) Not in database
$12$ 12.0.1194393600000000.1 \(\Z/6\Z \times \Z/6\Z\) Not in database
$12$ 12.0.1194393600000000.3 \(\Z/3\Z \times \Z/12\Z\) Not in database
$12$ 12.0.18662400000000.1 \(\Z/3\Z \times \Z/12\Z\) Not in database
$16$ 16.0.149587343098087735296.14 \(\Z/4\Z \times \Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/24\Z\) Not in database
$16$ 16.0.891610044825600000000.6 \(\Z/2\Z \times \Z/24\Z\) Not in database
$18$ 18.6.4663277994109219268198400000000.1 \(\Z/2\Z \times \Z/18\Z\) Not in database
$18$ 18.6.172713999781822935859200000000.1 \(\Z/36\Z\) Not in database
$18$ 18.6.9107964832244568883200000000.1 \(\Z/36\Z\) 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.