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This is the minimal-conductor curve with torsion $(\Z/2\Z) \times (\Z/6\Z)$.

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -19, 26]) # or

sage: E = EllipticCurve("30a2")

gp: E = ellinit([1, 0, 1, -19, 26]) \\ or

gp: E = ellinit("30a2")

magma: E := EllipticCurve([1, 0, 1, -19, 26]); // or

magma: E := EllipticCurve("30a2");

$$y^2 + x y + y = x^{3} - 19 x + 26$$

## Mordell-Weil group structure

$$\Z/{2}\Z \times \Z/{6}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(3, -2\right)$$, $$\left(-2, 8\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-5, 2\right)$$, $$\left(-2, 8\right)$$, $$\left(-2, -7\right)$$, $$\left(1, 2\right)$$, $$\left(1, -4\right)$$, $$\left(3, -2\right)$$, $$\left(4, 2\right)$$, $$\left(4, -7\right)$$, $$\left(13, 38\right)$$, $$\left(13, -52\right)$$

## 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: $$72900$$ = $$2^{2} \cdot 3^{6} \cdot 5^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{702595369}{72900}$$ = $$2^{-2} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{3} \cdot 127^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$3.35194825924$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$24$$  = $$2\cdot( 2 \cdot 3 )\cdot2$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$12$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form30.2.a.a

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: 4 $$\Gamma_0(N)$$-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);

$$L(E,1)$$ ≈ $$0.558658043207$$

## Local data

This elliptic curve is semistable.

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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$3$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2

## 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 X8c.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 6 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 5 \end{array}\right)$ 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
$$2$$ Cs
$$3$$ 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 and 6.
Its isogeny class 30.a 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
4 $$\Q(\sqrt{-2}, \sqrt{-3})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{3}, \sqrt{-5})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{2}, \sqrt{5})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
6 6.0.270000.1 $$\Z/6\Z \times \Z/6\Z$$ 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.

This is the curve of minimal conductor and torsion $(\Z/2\Z) \times (\Z/6\Z)$. Every elliptic curve $E/\Q$ with this torsion group must have conductor divisible by $30$ (for instance, if $E$ had good reduction at $5$ then the reduction mod $5$ would have at least $12$ points, which exceeds the Weil bound $(\sqrt5+1)^2 < 11$.