Properties

Label 6720.c1
Conductor $6720$
Discriminant $313528320000$
j-invariant \( \frac{229625675762164624948320008}{9568125} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-x^2-408240001x-3174701034815\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-x^2z-408240001xz^2-3174701034815z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-33067440108x-2314456256700432\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 0, -408240001, -3174701034815])
 
gp: E = ellinit([0, -1, 0, -408240001, -3174701034815])
 
magma: E := EllipticCurve([0, -1, 0, -408240001, -3174701034815]);
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 

Mordell-Weil group structure

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

Torsion generators

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

\( \left(-11665, 0\right) \) Copy content Toggle raw display

Integral points

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

\( \left(-11665, 0\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 6720 \)  =  $2^{6} \cdot 3 \cdot 5 \cdot 7$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $313528320000 $  =  $2^{15} \cdot 3^{7} \cdot 5^{4} \cdot 7 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{229625675762164624948320008}{9568125} \)  =  $2^{3} \cdot 3^{-7} \cdot 5^{-4} \cdot 7^{-1} \cdot 306180001^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $3.1065611481678616718814858284\dots$
Stable Faltings height: $2.2401271724679300351099456766\dots$

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: $0.033586957559078800673638355712\dots$
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: $ 4 $  = $ 2\cdot1\cdot2\cdot1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $2$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $36$ = $6^2$ (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) $ ≈ $ 1.2091304721268368242509808056 $

Modular invariants

Modular form   6720.2.a.c

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);
 

\( q - q^{3} - q^{5} - q^{7} + q^{9} - 4 q^{11} + 6 q^{13} + q^{15} + 6 q^{17} - 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 860160
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 4 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$(j)_{-}$
$2$ $2$ $I_{5}^{*}$ Additive -1 6 15 0
$3$ $1$ $I_{7}$ Non-split multiplicative 1 1 7 7
$5$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$7$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

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 8.12.0.12
sage: gens = [[1, 0, 8, 1], [1, 8, 0, 1], [64, 3, 61, 2], [7, 6, 162, 163], [161, 8, 160, 9], [113, 108, 110, 23], [124, 1, 167, 6], [1, 4, 4, 17], [104, 13, 143, 114]]
 
sage: GL(2,Integers(168)).subgroup(gens)
 
magma: Gens := [[1, 0, 8, 1], [1, 8, 0, 1], [64, 3, 61, 2], [7, 6, 162, 163], [161, 8, 160, 9], [113, 108, 110, 23], [124, 1, 167, 6], [1, 4, 4, 17], [104, 13, 143, 114]];
 
magma: sub<GL(2,Integers(168))|Gens>;
 

The image of the adelic Galois representation has level $168$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 64 & 3 \\ 61 & 2 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 162 & 163 \end{array}\right),\left(\begin{array}{rr} 161 & 8 \\ 160 & 9 \end{array}\right),\left(\begin{array}{rr} 113 & 108 \\ 110 & 23 \end{array}\right),\left(\begin{array}{rr} 124 & 1 \\ 167 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 104 & 13 \\ 143 & 114 \end{array}\right)$

$p$-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
Reduction type add nonsplit nonsplit nonsplit
$\lambda$-invariant(s) - 2 0 0
$\mu$-invariant(s) - 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 and 4.
Its isogeny class 6720.c consists of 4 curves linked by isogenies of degrees dividing 4.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{42}) \) \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{-3}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{-14}) \) \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-3}, \sqrt{-14})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ 8.4.1438916737499136.15 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ 8.0.149824733184.6 \(\Z/8\Z\) Not in database
$8$ 8.0.4441101041664.11 \(\Z/8\Z\) Not in database
$8$ deg 8 \(\Z/6\Z\) Not in database
$16$ deg 16 \(\Z/4\Z \oplus \Z/4\Z\) Not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$16$ deg 16 \(\Z/12\Z\) Not in database
$16$ deg 16 \(\Z/12\Z\) 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.