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Properties

Label 6930h3
Conductor $6930$
Discriminant $-2.283\times 10^{12}$
j-invariant \( -\frac{80677568161}{3131816380} \)
CM no
Rank $0$
Torsion structure \(\Z/{6}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3-x^2-810x+73440\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-x^2z-810xz^2+73440z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-12963x+4687198\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([1, -1, 0, -810, 73440])
 
gp: E = ellinit([1, -1, 0, -810, 73440])
 
magma: E := EllipticCurve([1, -1, 0, -810, 73440]);
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 

Mordell-Weil group structure

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

Torsion generators

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

\( \left(78, 654\right) \) Copy content Toggle raw display

Integral points

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

\( \left(-48, 24\right) \), \( \left(1, 269\right) \), \( \left(1, -270\right) \), \( \left(78, 654\right) \), \( \left(78, -732\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 6930 \)  =  $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-2283094141020 $  =  $-1 \cdot 2^{2} \cdot 3^{6} \cdot 5 \cdot 7^{6} \cdot 11^{3} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{80677568161}{3131816380} \)  =  $-1 \cdot 2^{-2} \cdot 5^{-1} \cdot 7^{-6} \cdot 11^{-3} \cdot 29^{3} \cdot 149^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.0515502659966922296004024199\dots$
Stable Faltings height: $0.50224412166263738390277980144\dots$

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: $0.68202081495670882708340611232\dots$
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: $ 72 $  = $ 2\cdot2\cdot1\cdot( 2 \cdot 3 )\cdot3 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $6$
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) $ ≈ $ 1.3640416299134176541668122246 $

Modular invariants

Modular form   6930.2.a.f

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);
 

\( q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} + q^{10} + q^{11} + 2 q^{13} - q^{14} + q^{16} - 6 q^{17} + 2 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 13824
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 5 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
$3$ $2$ $I_0^{*}$ Additive -1 2 6 0
$5$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$7$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$11$ $3$ $I_{3}$ Split multiplicative -1 1 3 3

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
$2$ 2B 2.3.0.1
$3$ 3B.1.1 3.8.0.1
sage: gens = [[3079, 9228, 7700, 9239], [4239, 2698, 6566, 5021], [4621, 12, 6, 73], [11, 2, 9190, 9231], [6730, 3, 5853, 9232], [5281, 12, 3966, 73], [1, 0, 12, 1], [9229, 12, 9228, 13], [1, 6, 6, 37], [1, 12, 0, 1], [10, 3, 1821, 9232]]
 
sage: GL(2,Integers(9240)).subgroup(gens)
 
magma: Gens := [[3079, 9228, 7700, 9239], [4239, 2698, 6566, 5021], [4621, 12, 6, 73], [11, 2, 9190, 9231], [6730, 3, 5853, 9232], [5281, 12, 3966, 73], [1, 0, 12, 1], [9229, 12, 9228, 13], [1, 6, 6, 37], [1, 12, 0, 1], [10, 3, 1821, 9232]];
 
magma: sub<GL(2,Integers(9240))|Gens>;
 

The image of the adelic Galois representation has level $9240$, index $96$, genus $1$, and generators

$\left(\begin{array}{rr} 3079 & 9228 \\ 7700 & 9239 \end{array}\right),\left(\begin{array}{rr} 4239 & 2698 \\ 6566 & 5021 \end{array}\right),\left(\begin{array}{rr} 4621 & 12 \\ 6 & 73 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 9190 & 9231 \end{array}\right),\left(\begin{array}{rr} 6730 & 3 \\ 5853 & 9232 \end{array}\right),\left(\begin{array}{rr} 5281 & 12 \\ 3966 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 9229 & 12 \\ 9228 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 3 \\ 1821 & 9232 \end{array}\right)$

$p$-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 7 11
Reduction type nonsplit add nonsplit split split
$\lambda$-invariant(s) 4 - 0 1 1
$\mu$-invariant(s) 0 - 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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, 3 and 6.
Its isogeny class 6930h consists of 4 curves linked by isogenies of degrees dividing 6.

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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-55}) \) \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$4$ 4.2.1552320.1 \(\Z/12\Z\) Not in database
$6$ 6.0.21870000.2 \(\Z/3\Z \oplus \Z/6\Z\) Not in database
$8$ deg 8 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$8$ 8.0.7289334581760000.309 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$9$ 9.3.3322925942906347470000.1 \(\Z/18\Z\) Not in database
$12$ deg 12 \(\Z/6\Z \oplus \Z/6\Z\) Not in database
$16$ deg 16 \(\Z/24\Z\) Not in database
$18$ 18.0.45927140031672784744455457990611433118437500000000.1 \(\Z/2\Z \oplus \Z/18\Z\) 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.