# Properties

 Label 30.a6 Conductor $30$ Discriminant $72900$ j-invariant $$\frac{702595369}{72900}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z \oplus \Z/{6}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

This is the minimal-conductor elliptic curve over $\Q$ with torsion subgroup $\Z/2\Z \times \Z/6\Z$.

## Simplified equation

 $$y^2+xy+y=x^3-19x+26$$ y^2+xy+y=x^3-19x+26 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-19xz^2+26z^3$$ y^2z+xyz+yz^2=x^3-19xz^2+26z^3 (dehomogenize, simplify) $$y^2=x^3-24003x+1296702$$ y^2=x^3-24003x+1296702 (homogenize, minimize)

comment: Define the curve

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

gp: E = ellinit([1, 0, 1, -19, 26])

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

oscar: E = EllipticCurve([1, 0, 1, -19, 26])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

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

magma: MordellWeilGroup(E);

## Torsion generators

$$\left(3, -2\right)$$, $$\left(1, 2\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

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

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$30$$ = $2 \cdot 3 \cdot 5$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $72900$ = $2^{2} \cdot 3^{6} \cdot 5^{2}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{702595369}{72900}$$ = $2^{-2} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{3} \cdot 127^{3}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.33171374939767582011699635763\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.33171374939767582011699635763\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.0045688963827404\dots$ Szpiro ratio: $5.989153060145781\dots$

## BSD invariants

 Analytic rank: $0$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $3.3519482592414964494482281339\dots$ comment: Real Period  sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Tamagawa product: $24$  = $2\cdot( 2 \cdot 3 )\cdot2$ comment: Tamagawa numbers  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E);  oscar: tamagawa_numbers(E) Torsion order: $12$ comment: Torsion order  sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E));  oscar: prod(torsion_structure(E)[1]) Analytic order of Ш: $1$ ( exact) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L(E,1)$ ≈ $0.55865804320691607490803802232$ comment: Special L-value  r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: [r,L1r] = ellanalyticrank(E); L1r/r!  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

## BSD formula

$\displaystyle 0.558658043 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 3.351948 \cdot 1.000000 \cdot 24}{12^2} \approx 0.558658043$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;

Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();

omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();

assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */

E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;

sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);

reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);

assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);

## Modular invariants

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

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ actual modular form, use for small N

[mf,F] = mffromell(E)

Ser(mfcoefs(mf,20),q)

\\ or just the series

Ser(ellan(E,20),q)*q

magma: ModularForm(E);

Modular degree: 4
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: no
Manin constant: 1
comment: Manin constant

magma: ManinConstant(E);

## Local data

This elliptic curve is semistable. There are 3 primes of bad reduction:

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

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)[5]

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

oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]

## Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2Cs 8.12.0.1
$3$ 3B.1.1 3.8.0.1

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

gens = [[31, 12, 66, 73], [67, 6, 114, 115], [109, 12, 108, 13], [97, 6, 0, 1], [9, 4, 104, 113], [81, 4, 22, 9], [1, 0, 12, 1], [1, 12, 0, 1]]

GL(2,Integers(120)).subgroup(gens)

Gens := [[31, 12, 66, 73], [67, 6, 114, 115], [109, 12, 108, 13], [97, 6, 0, 1], [9, 4, 104, 113], [81, 4, 22, 9], [1, 0, 12, 1], [1, 12, 0, 1]];

sub<GL(2,Integers(120))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$120 = 2^{3} \cdot 3 \cdot 5$$, index $384$, genus $5$, and generators

$\left(\begin{array}{rr} 31 & 12 \\ 66 & 73 \end{array}\right),\left(\begin{array}{rr} 67 & 6 \\ 114 & 115 \end{array}\right),\left(\begin{array}{rr} 109 & 12 \\ 108 & 13 \end{array}\right),\left(\begin{array}{rr} 97 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 104 & 113 \end{array}\right),\left(\begin{array}{rr} 81 & 4 \\ 22 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[120])$ is a degree-$92160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.

## Isogenies

gp: ellisomat(E)

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.

## Twists

This elliptic curve is its own minimal quadratic twist.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{6}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $4$ $$\Q(\sqrt{3}, \sqrt{-5})$$ $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $4$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{5})$$ $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $6$ 6.0.270000.1 $$\Z/6\Z \oplus \Z/6\Z$$ Not in database $9$ 9.3.143489070000.1 $$\Z/2\Z \oplus \Z/18\Z$$ Not in database $12$ 12.0.1194393600000000.1 $$\Z/6\Z \oplus \Z/12\Z$$ Not in database $16$ 16.0.11007531417600000000.1 $$\Z/4\Z \oplus \Z/12\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/24\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/24\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/24\Z$$ 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.

## 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.

## $p$-adic regulators

All $p$-adic regulators are identically $1$ since the rank is $0$.

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 $(\sqrt5+1)^2 < 11$.