Properties

Label 30a2
Conductor $30$
Discriminant $72900$
j-invariant \( \frac{702595369}{72900} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z \oplus \Z/{6}\Z\)

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

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3-19x+26\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-19xz^2+26z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-24003x+1296702\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([1, 0, 1, -19, 26])
 
gp: E = ellinit([1, 0, 1, -19, 26])
 
magma: E := EllipticCurve([1, 0, 1, -19, 26]);
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 

Mordell-Weil group structure

\(\Z/{2}\Z \oplus \Z/{6}\Z\)

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(3, -2\right) \), \( \left(1, 2\right) \) Copy content Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(-5, 2\right) \), \( \left(-2, 8\right) \), \( \left(-2, -7\right) \), \( \left(1, 2\right) \), \( \left(1, -4\right) \), \( \left(3, -2\right) \), \( \left(4, 2\right) \), \( \left(4, -7\right) \), \( \left(13, 38\right) \), \( \left(13, -52\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
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: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $-0.33171374939767582011699635763\dots$
Stable Faltings height: $-0.33171374939767582011699635763\dots$

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: if(E.disc>0,2,1)*E.omega[1]
 
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Real period: $3.3519482592414964494482281339\dots$
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: $ 24 $  = $ 2\cdot( 2 \cdot 3 )\cdot2 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $12$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
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);
 
Special value: $ L(E,1) $ ≈ $ 0.55865804320691607490803802232 $

Modular invariants

Modular form   30.2.a.a

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);
 

\( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + 2 q^{13} + 4 q^{14} - q^{15} + q^{16} + 6 q^{17} - q^{18} - 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

For more coefficients, see the Downloads section to the right.

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 4
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is semistable. There are 3 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$(j)_{-}$
$2$ $2$ $I_{2}$ 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

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 $\ell$-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
sage: 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]]
 
sage: GL(2,Integers(120)).subgroup(gens)
 
magma: 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]];
 
magma: sub<GL(2,Integers(120))|Gens>;
 

The image of the adelic Galois representation has level $120$, index $384$, genus $5$, and generators

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

$p$-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

$p$ 2 3 5
Reduction type nonsplit split nonsplit
$\lambda$-invariant(s) 0 1 0
$\mu$-invariant(s) 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 30a consists of 8 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$4$ \(\Q(\sqrt{3}, \sqrt{-5})\) \(\Z/2\Z \oplus \Z/12\Z\) 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.

Additional information

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