Properties

Label 39270cn3
Conductor 39270
Discriminant 16113547119140625000000000000
j-invariant \( \frac{15637378471582822120727563649467969}{16113547119140625000000000000} \)
CM no
Rank 0
Torsion Structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -5209708356, -144604741703664]); // or
magma: E := EllipticCurve("39270cn3");
sage: E = EllipticCurve([1, 0, 0, -5209708356, -144604741703664]) # or
sage: E = EllipticCurve("39270cn3")
gp: E = ellinit([1, 0, 0, -5209708356, -144604741703664]) \\ or
gp: E = ellinit("39270cn3")

\( y^2 + x y = x^{3} - 5209708356 x - 144604741703664 \)

Mordell-Weil group structure

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

Torsion generators

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

\( \left(83336, -41668\right) \)

Integral points

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

\( \left(83336, -41668\right) \)

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 39270 \)  =  \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 17\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(16113547119140625000000000000 \)  =  \(2^{12} \cdot 3 \cdot 5^{24} \cdot 7^{6} \cdot 11 \cdot 17 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{15637378471582822120727563649467969}{16113547119140625000000000000} \)  =  \(2^{-12} \cdot 3^{-1} \cdot 5^{-24} \cdot 7^{-6} \cdot 11^{-1} \cdot 17^{-1} \cdot 157^{3} \cdot 191^{3} \cdot 433^{3} \cdot 19259^{3}\)
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: \(0.0177714584457\)
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: \( 144 \)  = \( ( 2^{2} \cdot 3 )\cdot1\cdot2\cdot( 2 \cdot 3 )\cdot1\cdot1 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
Torsion order: \(2\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Analytic order of Ш: \(9\) (exact)

Modular invariants

Modular form 39270.2.a.cp

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)

\( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} + q^{11} + q^{12} + 2q^{13} + q^{14} - q^{15} + q^{16} + q^{17} + q^{18} + 8q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 55738368
\( \Gamma_0(N) \)-optimal: no
Manin constant: not computed

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])

\( L(E,1) \) ≈ \( 5.75795253642 \)

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\((j)_{-}\)
\(2\) \(12\) \( I_{12} \) Split multiplicative -1 1 12 12
\(3\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(5\) \(2\) \( I_{24} \) Non-split multiplicative 1 1 24 24
\(7\) \(6\) \( I_{6} \) Split multiplicative -1 1 6 6
\(11\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(17\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1

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 X13c.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 4 & 1 \end{array}\right)$ 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
\(2\) B
\(3\) B.1.2

$p$-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 11 17
Reduction type split split nonsplit split split split
$\lambda$-invariant(s) 5 5 0 1 3 1
$\mu$-invariant(s) 0 1 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, 4, 6 and 12.
Its isogeny class 39270cn 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 \(\Q(\sqrt{33}) \) \(\Z/4\Z\) Not in database
\(\Q(\sqrt{561}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
\(\Q(\sqrt{-3}) \) \(\Z/6\Z\) Not in database
\(\Q(\sqrt{17}) \) \(\Z/4\Z\) Not in database
3 3.1.8497467.4 \(\Z/6\Z\) Not in database
4 \(\Q(\sqrt{-3}, \sqrt{17})\) \(\Z/12\Z\) Not in database
\(\Q(\sqrt{17}, \sqrt{33})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{-3}, \sqrt{-11})\) \(\Z/12\Z\) Not in database
\(\Q(\sqrt{-3}, \sqrt{-187})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
6 \(x^{6} \) \(\mathstrut -\mathstrut 3 x^{5} \) \(\mathstrut -\mathstrut 21 x^{4} \) \(\mathstrut -\mathstrut 1075 x^{3} \) \(\mathstrut +\mathstrut 1851 x^{2} \) \(\mathstrut -\mathstrut 28803 x \) \(\mathstrut +\mathstrut 328234 \) \(\Z/12\Z\) Not in database
6.2.40508096378425929.1 \(\Z/2\Z \times \Z/6\Z\) Not in database
\(x^{6} \) \(\mathstrut -\mathstrut 3 x^{5} \) \(\mathstrut -\mathstrut 9 x^{4} \) \(\mathstrut -\mathstrut 1099 x^{3} \) \(\mathstrut +\mathstrut 1719 x^{2} \) \(\mathstrut -\mathstrut 15195 x \) \(\mathstrut +\mathstrut 321950 \) \(\Z/12\Z\) Not in database
6.0.216620836248267.2 \(\Z/3\Z \times \Z/6\Z\) 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.