# Properties

 Label 39270.cp3 Conductor 39270 Discriminant -66004356107812185925891924125000 j-invariant $$-\frac{63955658296770964115513956628279467969}{66004356107812185925891924125000}$$ CM no Rank 0 Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -83314083356, -9264307785578664]); // or
magma: E := EllipticCurve("39270cn7");
sage: E = EllipticCurve([1, 0, 0, -83314083356, -9264307785578664]) # or
sage: E = EllipticCurve("39270cn7")
gp: E = ellinit([1, 0, 0, -83314083356, -9264307785578664]) \\ or
gp: E = ellinit("39270cn7")

$$y^2 + x y = x^{3} - 83314083356 x - 9264307785578664$$

## Mordell-Weil group structure

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

## Torsion generators

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

$$\left(\frac{1333311}{4}, -\frac{1333311}{8}\right)$$

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()
None

## 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: $$-66004356107812185925891924125000$$ = $$-1 \cdot 2^{3} \cdot 3 \cdot 5^{6} \cdot 7^{24} \cdot 11 \cdot 17^{4}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$-\frac{63955658296770964115513956628279467969}{66004356107812185925891924125000}$$ = $$-1 \cdot 2^{-3} \cdot 3^{-1} \cdot 5^{-6} \cdot 7^{-24} \cdot 11^{-1} \cdot 13^{3} \cdot 17^{-4} \cdot 59^{3} \cdot 5213919167^{3}$$ 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: $$0.00444286461144$$ 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: $$576$$  = $$3\cdot1\cdot2\cdot( 2^{3} \cdot 3 )\cdot1\cdot2^{2}$$ 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)

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 222953472 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### 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$$(j)_{-}$$
$$2$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$3$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$5$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$7$$ $$24$$ $$I_{24}$$ Split multiplicative -1 1 24 24
$$11$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$17$$ $$4$$ $$I_{4}$$ 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.

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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 17 split split nonsplit split split split 5 5 0 1 3 1 2 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 39270.cp 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 $$\Q(\sqrt{2})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{-66})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$$\Q(\sqrt{-33})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{-3})$$ $$\Z/6\Z$$ Not in database
3 3.1.8497467.4 $$\Z/6\Z$$ Not in database
4 $$\Q(\sqrt{-3}, \sqrt{11})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{-3}, \sqrt{22})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
$$\Q(\sqrt{2}, \sqrt{-3})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{2}, \sqrt{-33})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
6 $$x^{6}$$ $$\mathstrut +\mathstrut 99 x^{4}$$ $$\mathstrut -\mathstrut 53955 x^{2}$$ $$\mathstrut +\mathstrut 3421572$$ $$\Z/12\Z$$ Not in database
$$x^{6}$$ $$\mathstrut +\mathstrut 198 x^{4}$$ $$\mathstrut -\mathstrut 1122 x^{3}$$ $$\mathstrut +\mathstrut 13068 x^{2}$$ $$\mathstrut +\mathstrut 222156 x$$ $$\mathstrut +\mathstrut 602217$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
6.0.216620836248267.2 $$\Z/3\Z \times \Z/6\Z$$ Not in database
$$x^{6}$$ $$\mathstrut -\mathstrut 6 x^{4}$$ $$\mathstrut -\mathstrut 1122 x^{3}$$ $$\mathstrut +\mathstrut 12 x^{2}$$ $$\mathstrut -\mathstrut 6732 x$$ $$\mathstrut +\mathstrut 314713$$ $$\Z/12\Z$$ 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.