Properties

Label 419430bm5
Conductor $419430$
Discriminant $-3.264\times 10^{32}$
j-invariant \( -\frac{1109449359233669878543860978690109441}{326424055277608242539062500000000} \)
CM no
Rank $0$
Torsion structure \(\Z/{8}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3+x^2-21567240960x+1497262071813537\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3+x^2z-21567240960xz^2+1497262071813537z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-27951144284187x+69856678489696653366\) Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([1, 1, 1, -21567240960, 1497262071813537])
 
Copy content gp:E = ellinit([1, 1, 1, -21567240960, 1497262071813537])
 
Copy content magma:E := EllipticCurve([1, 1, 1, -21567240960, 1497262071813537]);
 
Copy content oscar:E = elliptic_curve([1, 1, 1, -21567240960, 1497262071813537])
 
Copy content comment:Simplified equation
 
Copy content sage:E.short_weierstrass_model()
 
Copy content magma:WeierstrassModel(E);
 
Copy content oscar:short_weierstrass_model(E)
 

Mordell-Weil group structure

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

Copy content comment:Mordell-Weil group
 
Copy content magma:MordellWeilGroup(E);
 

Mordell-Weil generators

$P$$\hat{h}(P)$Order
\( \left(-13583, 42288041\right) \)$0$$8$

$P$$\hat{h}(P)$Order
\([-13583:42288041:1]\)$0$$8$

$P$$\hat{h}(P)$Order
\( \left(-488973, 9132750000\right) \)$0$$8$

Integral points

\( \left(-13583, 42288041\right) \), \( \left(-13583, -42274459\right) \), \( \left(88917, 16765541\right) \), \( \left(88917, -16854459\right) \), \( \left(257017, 113591141\right) \), \( \left(257017, -113848159\right) \) Copy content Toggle raw display

\([-13583:42288041:1]\), \([-13583:-42274459:1]\), \([88917:16765541:1]\), \([88917:-16854459:1]\), \([257017:113591141:1]\), \([257017:-113848159:1]\) Copy content Toggle raw display

\((-488973,\pm 9132750000)\), \((3201027,\pm 3630960000)\), \((9252627,\pm 24563444400)\) Copy content Toggle raw display

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

Invariants

Conductor: $N$  =  \( 419430 \) = $2 \cdot 3 \cdot 5 \cdot 11 \cdot 31 \cdot 41$
Copy content comment:Conductor
 
Copy content sage:E.conductor().factor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Copy content oscar:conductor(E)
 
Minimal Discriminant: $\Delta$  =  $-326424055277608242539062500000000$ = $-1 \cdot 2^{8} \cdot 3^{2} \cdot 5^{16} \cdot 11^{2} \cdot 31^{2} \cdot 41^{8} $
Copy content comment:Discriminant
 
Copy content sage:E.discriminant().factor()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Copy content oscar:discriminant(E)
 
j-invariant: $j$  =  \( -\frac{1109449359233669878543860978690109441}{326424055277608242539062500000000} \) = $-1 \cdot 2^{-8} \cdot 3^{-2} \cdot 5^{-16} \cdot 11^{-2} \cdot 31^{-2} \cdot 41^{-8} \cdot 263^{3} \cdot 2999^{3} \cdot 1312513^{3}$
Copy content comment:j-invariant
 
Copy content sage:E.j_invariant().factor()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Copy content oscar:j_invariant(E)
 
Endomorphism ring: $\mathrm{End}(E)$ = $\Z$
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$  =  \(\Z\)    (no potential complex multiplication)
Copy content comment:Potential complex multiplication
 
Copy content sage:E.has_cm()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$
Faltings height: $h_{\mathrm{Faltings}}$ ≈ $4.9533361837048271130920690435$
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $4.9533361837048271130920690435$
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $1.0132622478122608$
Szpiro ratio: $\sigma_{m}$ ≈ $6.442436787493309$
Intrinsic torsion order: $\#E(\mathbb Q)_\text{tors}^\text{is}$ = $1$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$ = $ 0$
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: $r$ = $ 0$
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: $\mathrm{Reg}(E/\Q)$ = $1$
Copy content comment:Regulator
 
Copy content sage:E.regulator()
 
Copy content gp:G = E.gen \\ if available matdet(ellheightmatrix(E,G))
 
Copy content magma:Regulator(E);
 
Real period: $\Omega$ ≈ $0.016245640451975917948134667366$
Copy content comment:Real Period
 
Copy content sage:E.period_lattice().omega()
 
Copy content gp:if(E.disc>0,2,1)*E.omega[1]
 
Copy content magma:(Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: $\prod_{p}c_p$ = $ 8192 $  = $ 2^{3}\cdot2\cdot2^{4}\cdot2\cdot2\cdot2^{3} $
Copy content comment:Tamagawa numbers
 
Copy content sage:E.tamagawa_numbers()
 
Copy content gp:gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Copy content magma:TamagawaNumbers(E);
 
Copy content oscar:tamagawa_numbers(E)
 
Torsion order: $\#E(\Q)_{\mathrm{tor}}$ = $8$
Copy content comment:Torsion order
 
Copy content sage:E.torsion_order()
 
Copy content gp:elltors(E)[1]
 
Copy content magma:Order(TorsionSubgroup(E));
 
Copy content oscar:prod(torsion_structure(E)[1])
 
Special value: $ L(E,1)$ ≈ $2.0794419778529174973612374228 $
Copy content comment:Special L-value
 
Copy content sage:r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
Copy content gp:[r,L1r] = ellanalyticrank(E); L1r/r!
 
Copy content magma:Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Ш${}_{\mathrm{an}}$  =  $1$    (exact)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

$$\begin{aligned} 2.079441978 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.016246 \cdot 1.000000 \cdot 8192}{8^2} \\ & \approx 2.079441978\end{aligned}$$

Copy content comment:BSD formula
 
Copy content sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha) E = EllipticCurve([1, 1, 1, -21567240960, 1497262071813537]); 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)))
 
Copy content magma:/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */ E := EllipticCurve([1, 1, 1, -21567240960, 1497262071813537]); 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 419430.2.a.bm

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

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 

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

Modular degree: 1811939328
Copy content comment:Modular degree
 
Copy content sage:E.modular_degree()
 
Copy content gp:ellmoddegree(E)
 
Copy content magma:ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: no
Manin constant: 1 (conditional*)
Copy content comment:Manin constant
 
Copy content magma:ManinConstant(E);
 
* The Manin constant is correct provided that curve 419430bm1 is optimal.

Local data at primes of bad reduction

This elliptic curve is semistable. There are 6 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $8$ $I_{8}$ split multiplicative -1 1 8 8
$3$ $2$ $I_{2}$ nonsplit multiplicative 1 1 2 2
$5$ $16$ $I_{16}$ split multiplicative -1 1 16 16
$11$ $2$ $I_{2}$ nonsplit multiplicative 1 1 2 2
$31$ $2$ $I_{2}$ nonsplit multiplicative 1 1 2 2
$41$ $8$ $I_{8}$ split multiplicative -1 1 8 8

Copy content comment:Local data
 
Copy content sage:E.local_data()
 
Copy content gp:ellglobalred(E)[5]
 
Copy content magma:[LocalInformation(E,p) : p in BadPrimes(E)];
 
Copy content 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 $\ell$-adic index
$2$ 2B 8.96.0.99 $96$

Copy content comment:Mod p Galois image
 
Copy content sage:rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
Copy content magma:[GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

Copy content comment:Adelic image of Galois representation
 
Copy content sage:gens = [[13, 16, 1694, 5874105], [1, 0, 32, 1], [2440345, 16, 6709494, 6709993], [1, 32, 0, 1], [2618881, 8, 0, 1], [2236985, 16, 6709494, 6709993], [6710849, 32, 6710848, 33], [3463705, 32, 4982422, 4329], [25, 16, 6709144, 6709769], [1677739, 3355448, 5239975, 6290229], [1, 32, 4, 129], [1342177, 16, 0, 1]] GL(2,Integers(6710880)).subgroup(gens)
 
Copy content magma:Gens := [[13, 16, 1694, 5874105], [1, 0, 32, 1], [2440345, 16, 6709494, 6709993], [1, 32, 0, 1], [2618881, 8, 0, 1], [2236985, 16, 6709494, 6709993], [6710849, 32, 6710848, 33], [3463705, 32, 4982422, 4329], [25, 16, 6709144, 6709769], [1677739, 3355448, 5239975, 6290229], [1, 32, 4, 129], [1342177, 16, 0, 1]]; sub<GL(2,Integers(6710880))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6710880 = 2^{5} \cdot 3 \cdot 5 \cdot 11 \cdot 31 \cdot 41 \), index $768$, genus $13$, and generators

$\left(\begin{array}{rr} 13 & 16 \\ 1694 & 5874105 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 2440345 & 16 \\ 6709494 & 6709993 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2618881 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2236985 & 16 \\ 6709494 & 6709993 \end{array}\right),\left(\begin{array}{rr} 6710849 & 32 \\ 6710848 & 33 \end{array}\right),\left(\begin{array}{rr} 3463705 & 32 \\ 4982422 & 4329 \end{array}\right),\left(\begin{array}{rr} 25 & 16 \\ 6709144 & 6709769 \end{array}\right),\left(\begin{array}{rr} 1677739 & 3355448 \\ 5239975 & 6290229 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 4 & 129 \end{array}\right),\left(\begin{array}{rr} 1342177 & 16 \\ 0 & 1 \end{array}\right)$.

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

The torsion field $K:=\Q(E[6710880])$ is a degree-$383030783020892160000000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6710880\Z)$.

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 $4$ \( 1 \)
$3$ nonsplit multiplicative $4$ \( 139810 = 2 \cdot 5 \cdot 11 \cdot 31 \cdot 41 \)
$5$ split multiplicative $6$ \( 83886 = 2 \cdot 3 \cdot 11 \cdot 31 \cdot 41 \)
$11$ nonsplit multiplicative $12$ \( 38130 = 2 \cdot 3 \cdot 5 \cdot 31 \cdot 41 \)
$31$ nonsplit multiplicative $32$ \( 13530 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 41 \)
$41$ split multiplicative $42$ \( 10230 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 31 \)

Isogenies

Copy content comment:Isogenies
 
Copy content gp:ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 419430bm consists of 8 curves linked by isogenies of degrees dividing 16.

Twists

This elliptic curve is its own minimal quadratic twist.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

$p$-adic regulators

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