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Properties

Label 210.e6
Conductor $210$
Discriminant $51438240000$
j-invariant \( \frac{135487869158881}{51438240000} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z \oplus \Z/{8}\Z\)

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This is the minimal-conductor curve with torsion $(\Z/2\Z) \times (\Z/8\Z)$.

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3-1070x+7812\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-1070xz^2+7812z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-1386747x+368636886\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 0, 0, -1070, 7812])
 
gp: E = ellinit([1, 0, 0, -1070, 7812])
 
magma: E := EllipticCurve([1, 0, 0, -1070, 7812]);
 
oscar: E = elliptic_curve([1, 0, 0, -1070, 7812])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z/{2}\Z \oplus \Z/{8}\Z\)

magma: MordellWeilGroup(E);
 

Torsion generators

\( \left(-36, 18\right) \), \( \left(4, 58\right) \) Copy content Toggle raw display

comment: Torsion subgroup
 
sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 
oscar: torsion_structure(E)
 

Integral points

\( \left(-36, 18\right) \), \( \left(-26, 148\right) \), \( \left(-26, -122\right) \), \( \left(-8, 130\right) \), \( \left(-8, -122\right) \), \( \left(4, 58\right) \), \( \left(4, -62\right) \), \( \left(28, -14\right) \), \( \left(34, 88\right) \), \( \left(34, -122\right) \), \( \left(64, 418\right) \), \( \left(64, -482\right) \), \( \left(244, 3658\right) \), \( \left(244, -3902\right) \) Copy content Toggle raw display

comment: Integral points
 
sage: E.integral_points()
 
magma: IntegralPoints(E);
 

Invariants

Conductor: \( 210 \)  =  $2 \cdot 3 \cdot 5 \cdot 7$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $51438240000 $  =  $2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{135487869158881}{51438240000} \)  =  $2^{-8} \cdot 3^{-8} \cdot 5^{-4} \cdot 7^{-2} \cdot 51361^{3}$
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: $0.75470864605564508282719006522\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $0.75470864605564508282719006522\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $1.0190986885579905\dots$
Szpiro ratio: $6.085515026358872\dots$

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: $1.0259330100195332631677131886\dots$
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: $ 512 $  = $ 2^{3}\cdot2^{3}\cdot2^{2}\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: $16$
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) $ ≈ $ 2.0518660200390665263354263771 $
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

$\displaystyle 2.051866020 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.025933 \cdot 1.000000 \cdot 512}{16^2} \approx 2.051866020$

# 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

Modular form   210.2.a.e

\( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + q^{10} - 4 q^{11} + q^{12} - 2 q^{13} - q^{14} + q^{15} + q^{16} + 2 q^{17} + q^{18} + 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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);
 

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

Modular degree: 256
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 4 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $v_p(N)$ $v_p(\Delta)$ $v_p(\mathrm{den}(j))$
$2$ $8$ $I_{8}$ split multiplicative -1 1 8 8
$3$ $8$ $I_{8}$ split multiplicative -1 1 8 8
$5$ $4$ $I_{4}$ split multiplicative -1 1 4 4
$7$ $2$ $I_{2}$ nonsplit 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 8.96.0.40

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 = [[15, 16, 74, 289], [1121, 8, 0, 1], [1, 16, 0, 1], [1353, 16, 1076, 121], [13, 8, 320, 617], [241, 16, 1446, 97], [1, 0, 16, 1], [1, 16, 4, 65], [1665, 16, 1664, 17]]
 
GL(2,Integers(1680)).subgroup(gens)
 
Gens := [[15, 16, 74, 289], [1121, 8, 0, 1], [1, 16, 0, 1], [1353, 16, 1076, 121], [13, 8, 320, 617], [241, 16, 1446, 97], [1, 0, 16, 1], [1, 16, 4, 65], [1665, 16, 1664, 17]];
 
sub<GL(2,Integers(1680))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \), index $768$, genus $13$, and generators

$\left(\begin{array}{rr} 15 & 16 \\ 74 & 289 \end{array}\right),\left(\begin{array}{rr} 1121 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1353 & 16 \\ 1076 & 121 \end{array}\right),\left(\begin{array}{rr} 13 & 8 \\ 320 & 617 \end{array}\right),\left(\begin{array}{rr} 241 & 16 \\ 1446 & 97 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 1665 & 16 \\ 1664 & 17 \end{array}\right)$.

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

The torsion field $K:=\Q(E[1680])$ is a degree-$1486356480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1680\Z)$.

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$ Reduction type Serre weight Serre conductor
$2$ split multiplicative $4$ \( 1 \)
$3$ split multiplicative $4$ \( 70 = 2 \cdot 5 \cdot 7 \)
$5$ split multiplicative $6$ \( 42 = 2 \cdot 3 \cdot 7 \)
$7$ nonsplit multiplicative $8$ \( 30 = 2 \cdot 3 \cdot 5 \)

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 210.e consists of 8 curves linked by isogenies of degrees dividing 16.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{8}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$4$ \(\Q(i, \sqrt{7})\) \(\Z/4\Z \oplus \Z/8\Z\) not in database
$4$ \(\Q(\sqrt{-6}, \sqrt{10})\) \(\Z/2\Z \oplus \Z/16\Z\) not in database
$4$ \(\Q(\sqrt{6}, \sqrt{70})\) \(\Z/2\Z \oplus \Z/16\Z\) not in database
$8$ 8.2.4253299470000.8 \(\Z/2\Z \oplus \Z/24\Z\) not in database
$16$ deg 16 \(\Z/8\Z \oplus \Z/8\Z\) not in database
$16$ 16.0.5951500145509072896.2 \(\Z/4\Z \oplus \Z/16\Z\) not in database
$16$ 16.0.63456228123711897600000000.20 \(\Z/4\Z \oplus \Z/16\Z\) not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/32\Z\) not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/32\Z\) 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

$p$ 2 3 5 7
Reduction type split split split nonsplit
$\lambda$-invariant(s) 2 5 1 0
$\mu$-invariant(s) 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

$p$-adic regulators

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

Additional information

This is the curve of minimal conductor and torsion $(\Z/2\Z) \times (\Z/8\Z)$. Every elliptic curve $E/\Q$ with this torsion group must have conductor divisible by $210 = 2 \cdot 3 \cdot 5 \cdot 7$ (for instance, if $E$ had good reduction at $7$ then the reduction mod $7$ would have at least $16$ points, which exceeds the Weil bound $(\sqrt7+1)^2 < 14$.