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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)  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  magma: RealPeriod(E); Real period: $$3.3519482592414964494482281339$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$6$$  = $$1\cdot3\cdot2$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$6$$ 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);

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

 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/factorial(ar)

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)

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

$p$ Reduction type $\lambda$-invariant(s) 2 3 5 nonsplit split nonsplit 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 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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{6})$$ $$\Z/2\Z \times \Z/6\Z$$ 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.