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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -28, 272])

gp: E = ellinit([1, 0, 0, -28, 272])

magma: E := EllipticCurve([1, 0, 0, -28, 272]);

$$y^2+xy=x^3-28x+272$$ ## Mordell-Weil group structure

$$\Z/{10}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(2, 14\right)$$ ## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-8, 4\right)$$, $$\left(-4, 20\right)$$, $$\left(-4, -16\right)$$, $$\left(2, 14\right)$$, $$\left(2, -16\right)$$, $$\left(8, 20\right)$$, $$\left(8, -28\right)$$, $$\left(32, 164\right)$$, $$\left(32, -196\right)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$150$$ = $$2 \cdot 3 \cdot 5^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-31104000$$ = $$-1 \cdot 2^{10} \cdot 3^{5} \cdot 5^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{19465109}{248832}$$ = $$-1 \cdot 2^{-10} \cdot 3^{-5} \cdot 269^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$0.11944777360291933688272067076\dots$$ Stable Faltings height: $$-0.28291170450560575676746916255\dots$$

## 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  magma: RealPeriod(E); Real period: $$1.7692592482524138311369605540\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$100$$  = $$( 2 \cdot 5 )\cdot5\cdot2$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$10$$ 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)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q + q^{2} + q^{3} + q^{4} + q^{6} - 2q^{7} + q^{8} + q^{9} + 2q^{11} + q^{12} - 6q^{13} - 2q^{14} + q^{16} - 2q^{17} + q^{18} + O(q^{20})$$ For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 40 $$\Gamma_0(N)$$-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/factorial(ar)

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

$$L(E,1)$$ ≈ $$1.7692592482524138311369605540290421893$$

## Local data

This elliptic curve is not semistable. There are 3 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$10$$ $$I_{10}$$ Split multiplicative -1 1 10 10
$$3$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$5$$ $$2$$ $$III$$ Additive -1 2 3 0

## 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 X6.

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

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
$$5$$ B.1.1

## $p$-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) 2 3 5 split split add 1 1 - 0 0 -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

An entry - indicates that the invariants are not computed because the reduction is additive.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 5 and 10.
Its isogeny class 150.c consists of 4 curves linked by isogenies of degrees dividing 10.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{10}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-15})$$ $$\Z/2\Z \times \Z/10\Z$$ Not in database $4$ 4.2.24000.2 $$\Z/20\Z$$ Not in database $8$ 8.0.11664000000.8 $$\Z/2\Z \times \Z/20\Z$$ Not in database $8$ 8.0.5184000000.10 $$\Z/2\Z \times \Z/20\Z$$ Not in database $8$ 8.2.44286750000.2 $$\Z/30\Z$$ Not in database $16$ Deg 16 $$\Z/40\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/30\Z$$ Not in database $20$ 20.0.4656612873077392578125.1 $$\Z/5\Z \times \Z/10\Z$$ 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.