# Properties

 Label 930.o4 Conductor $930$ Discriminant $1.271\times 10^{12}$ j-invariant $$\frac{5601911201812801}{1271193750000}$$ CM no Rank $0$ Torsion structure $$\Z/{8}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -3700, 67232])

gp: E = ellinit([1, 0, 0, -3700, 67232])

magma: E := EllipticCurve([1, 0, 0, -3700, 67232]);

$$y^2+xy=x^3-3700x+67232$$

## Mordell-Weil group structure

$$\Z/{8}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(14, 128\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-46, 398\right)$$, $$\left(-46, -352\right)$$, $$\left(14, 128\right)$$, $$\left(14, -142\right)$$, $$\left(104, 848\right)$$, $$\left(104, -952\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$930$$ = $$2 \cdot 3 \cdot 5 \cdot 31$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$1271193750000$$ = $$2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 31$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{5601911201812801}{1271193750000}$$ = $$2^{-4} \cdot 3^{-8} \cdot 5^{-8} \cdot 31^{-1} \cdot 177601^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.0352520519348703797869572516\dots$$ Stable Faltings height: $$1.0352520519348703797869572516\dots$$

## 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: $$0.81094462040509483207370725743\dots$$ 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: $$256$$  = $$2^{2}\cdot2^{3}\cdot2^{3}\cdot1$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$8$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2048 $$\Gamma_0(N)$$-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);

$$L(E,1)$$ ≈ $$3.2437784816203793282948290297226753766$$

## Local data

This elliptic curve is semistable. There are 4 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$3$$ $$8$$ $$I_{8}$$ Split multiplicative -1 1 8 8
$$5$$ $$8$$ $$I_{8}$$ Split multiplicative -1 1 8 8
$$31$$ $$1$$ $$I_{1}$$ Non-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 X102p.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right)$ and has index 48.

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

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 31 split split split nonsplit 8 1 1 0 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 930.o consists of 4 curves linked by isogenies of degrees dividing 8.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{8}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{31})$$ $$\Z/2\Z \times \Z/8\Z$$ Not in database $4$ 4.0.446400.3 $$\Z/16\Z$$ Not in database $8$ 8.0.58163441238016.5 $$\Z/4\Z \times \Z/8\Z$$ Not in database $8$ 8.8.11502047705760000.3 $$\Z/2\Z \times \Z/16\Z$$ Not in database $8$ 8.0.3064021032960000.29 $$\Z/2\Z \times \Z/16\Z$$ Not in database $8$ 8.2.1635989745870000.11 $$\Z/24\Z$$ Not in database $16$ Deg 16 $$\Z/32\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/24\Z$$ 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.