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Properties

Label 411840hc6
Conductor $411840$
Discriminant $5.883\times 10^{24}$
j-invariant \( \frac{84415028961834287121601}{30783551683856400} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z \oplus \Z/{2}\Z\)

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Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-526406412x+4647217420784\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-526406412xz^2+4647217420784z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-526406412x+4647217420784\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, 0, 0, -526406412, 4647217420784])
 
gp: E = ellinit([0, 0, 0, -526406412, 4647217420784])
 
magma: E := EllipticCurve([0, 0, 0, -526406412, 4647217420784]);
 
oscar: E = EllipticCurve([0, 0, 0, -526406412, 4647217420784])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

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

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generator and height

$P$ =  \(\left(\frac{18935678}{961}, \frac{41333608960}{29791}\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $12.789387418612607468403442426$

sage: E.gens()
 
magma: Generators(E);
 
gp: E.gen
 

Torsion generators

\( \left(13054, 0\right) \), \( \left(13438, 0\right) \) 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

\( \left(-26492, 0\right) \), \( \left(13054, 0\right) \), \( \left(13438, 0\right) \) Copy content Toggle raw display

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

Invariants

Conductor: \( 411840 \)  =  $2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $5882828338634769196646400 $  =  $2^{22} \cdot 3^{8} \cdot 5^{2} \cdot 11^{6} \cdot 13^{6} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{84415028961834287121601}{30783551683856400} \)  =  $2^{-4} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{6} \cdot 11^{-6} \cdot 13^{-6} \cdot 31^{3} \cdot 28879^{3}$
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: $3.7211119804196985699649222415\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $2.1320850652457257601414514409\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $1.0472236692626555\dots$
Szpiro ratio: $5.5581827480840404\dots$

BSD invariants

Analytic rank: $1$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $12.789387418612607468403442426\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.074364288216282455412872727545\dots$
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: $ 128 $  = $ 2^{2}\cdot2^{2}\cdot2\cdot2\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: $4$
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: $1$ ( rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Special value: $ L'(E,1) $ ≈ $ 7.6085895368592369306827295768 $
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

$\displaystyle 7.608589537 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.074364 \cdot 12.789387 \cdot 128}{4^2} \approx 7.608589537$

# 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 411840.2.a.hc

\( q + q^{5} - 4 q^{7} - q^{11} - q^{13} + 6 q^{17} + 4 q^{19} + O(q^{20}) \) 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: 127401984
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: no
Manin constant: 1 (conditional*)
comment: Manin constant
 
magma: ManinConstant(E);
 
* The Manin constant is correct provided that curve 411840hc1 is optimal.

Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $4$ $I_{12}^{*}$ Additive 1 6 22 4
$3$ $4$ $I_{2}^{*}$ Additive -1 2 8 2
$5$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$11$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$13$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6

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 2.6.0.1
$3$ 3B 3.4.0.1

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 = [[15833, 17154, 15846, 8585], [15443, 17154, 0, 17159], [14293, 17154, 14306, 8585], [8591, 17148, 8592, 17147], [1, 0, 12, 1], [17151, 17156, 8596, 8587], [4297, 12, 144, 247], [17149, 12, 17148, 13], [1, 12, 0, 1], [7801, 6, 0, 1], [9, 4, 17144, 17153], [8579, 0, 0, 17159]]
 
GL(2,Integers(17160)).subgroup(gens)
 
Gens := [[15833, 17154, 15846, 8585], [15443, 17154, 0, 17159], [14293, 17154, 14306, 8585], [8591, 17148, 8592, 17147], [1, 0, 12, 1], [17151, 17156, 8596, 8587], [4297, 12, 144, 247], [17149, 12, 17148, 13], [1, 12, 0, 1], [7801, 6, 0, 1], [9, 4, 17144, 17153], [8579, 0, 0, 17159]];
 
sub<GL(2,Integers(17160))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 17160 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \), index $384$, genus $5$, and generators

$\left(\begin{array}{rr} 15833 & 17154 \\ 15846 & 8585 \end{array}\right),\left(\begin{array}{rr} 15443 & 17154 \\ 0 & 17159 \end{array}\right),\left(\begin{array}{rr} 14293 & 17154 \\ 14306 & 8585 \end{array}\right),\left(\begin{array}{rr} 8591 & 17148 \\ 8592 & 17147 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 17151 & 17156 \\ 8596 & 8587 \end{array}\right),\left(\begin{array}{rr} 4297 & 12 \\ 144 & 247 \end{array}\right),\left(\begin{array}{rr} 17149 & 12 \\ 17148 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7801 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 17144 & 17153 \end{array}\right),\left(\begin{array}{rr} 8579 & 0 \\ 0 & 17159 \end{array}\right)$.

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

The torsion field $K:=\Q(E[17160])$ is a degree-$31882346496000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/17160\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 411840hc consists of 8 curves linked by isogenies of degrees dividing 12.

Twists

The minimal quadratic twist of this elliptic curve is 4290bb6, its twist by $-24$.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

$p$-adic regulators

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.