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Properties

Label 406560ch4
Conductor $406560$
Discriminant $8.679\times 10^{15}$
j-invariant \( \frac{229625675762164624948320008}{9568125} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-x^2-12349260040x-528209408557400\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-x^2z-12349260040xz^2-528209408557400z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-1000290063267x-385067659708534374\) Copy content Toggle raw display (homogenize, minimize)

Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([0, -1, 0, -12349260040, -528209408557400])
 
Copy content gp:E = ellinit([0, -1, 0, -12349260040, -528209408557400])
 
Copy content magma:E := EllipticCurve([0, -1, 0, -12349260040, -528209408557400]);
 
Copy content oscar:E = elliptic_curve([0, -1, 0, -12349260040, -528209408557400])
 
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

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

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

Mordell-Weil generators

$P$$\hat{h}(P)$Order
\( \left(\frac{97316366793603325}{441795514329}, \frac{25328079777437844879188530}{293651317077656733}\right) \)$35.767394671076338021003630613$$\infty$
\( \left(-64159, 0\right) \)$0$$2$

$P$$\hat{h}(P)$Order
\([64683950731271877251025:25328079777437844879188530:293651317077656733]\)$35.767394671076338021003630613$$\infty$
\([-64159:0:1]\)$0$$2$

$P$$\hat{h}(P)$Order
\( \left(\frac{97316219528431882}{49088390481}, \frac{25328079777437844879188530}{10875974706579879}\right) \)$35.767394671076338021003630613$$\infty$
\( \left(-577434, 0\right) \)$0$$2$

Integral points

\( \left(-64159, 0\right) \) Copy content Toggle raw display

\([-64159:0:1]\) Copy content Toggle raw display

\( \left(-64159, 0\right) \) Copy content Toggle raw display

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

Invariants

Conductor: $N$  =  \( 406560 \) = $2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}$
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: $\Delta$  =  $8678664751680000$ = $2^{9} \cdot 3^{7} \cdot 5^{4} \cdot 7 \cdot 11^{6} $
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: $j$  =  \( \frac{229625675762164624948320008}{9568125} \) = $2^{3} \cdot 3^{-7} \cdot 5^{-4} \cdot 7^{-1} \cdot 306180001^{3}$
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: $h_{\mathrm{Faltings}}$ ≈ $3.9589351942870742892038415566$
Copy content comment:Faltings height
 
Copy content gp:ellheight(E)
 
Copy content magma:FaltingsHeight(E);
 
Copy content oscar:faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $2.2401271724679300351099456765$
Copy content comment:Stable Faltings height
 
Copy content magma:StableFaltingsHeight(E);
 
Copy content oscar:stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $1.0816058099434143$
Szpiro ratio: $\sigma_{m}$ ≈ $6.2966414079891555$
Intrinsic torsion order: $\#E(\mathbb Q)_\text{tors}^\text{is}$ = $$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$ = $ 1$
Copy content comment:Analytic rank
 
Copy content sage:E.analytic_rank()
 
Copy content gp:ellanalyticrank(E)
 
Copy content magma:AnalyticRank(E);
 
Mordell-Weil rank: $r$ = $ 1$
Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 
Regulator: $\mathrm{Reg}(E/\Q)$ ≈ $35.767394671076338021003630613$
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: $\Omega$ ≈ $0.014321526823602170336776376328$
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: $\prod_{p}c_p$ = $ 16 $  = $ 1\cdot1\cdot2^{2}\cdot1\cdot2^{2} $
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: $\#E(\Q)_{\mathrm{tor}}$ = $2$
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)$ ≈ $8.1958992350749616214898214348 $
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${}_{\mathrm{an}}$  ≈  $4$ = $2^2$    (rounded)
Copy content comment:Order of Sha
 
Copy content sage:E.sha().an_numerical()
 
Copy content magma:MordellWeilShaInformation(E);
 

BSD formula

$$\begin{aligned} 8.195899235 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.014322 \cdot 35.767395 \cdot 16}{2^2} \\ & \approx 8.195899235\end{aligned}$$

Copy content comment:BSD formula
 
Copy content sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha) E = EllipticCurve([0, -1, 0, -12349260040, -528209408557400]); 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([0, -1, 0, -12349260040, -528209408557400]); 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 406560.2.a.ch

\( q - q^{3} + q^{5} - q^{7} + q^{9} + 6 q^{13} - q^{15} - 6 q^{17} + 4 q^{19} + O(q^{20}) \) 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: 275251200
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 406560ch1 is optimal.

Local data at primes of bad reduction

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

$p$ Tamagawa number Kodaira symbol Reduction type Root number $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $1$ $I_0^{*}$ additive 1 5 9 0
$3$ $1$ $I_{7}$ nonsplit multiplicative 1 1 7 7
$5$ $4$ $I_{4}$ split multiplicative -1 1 4 4
$7$ $1$ $I_{1}$ nonsplit multiplicative 1 1 1 1
$11$ $4$ $I_0^{*}$ additive -1 2 6 0

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 4.6.0.1 $6$

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 = [[1, 0, 8, 1], [1, 8, 0, 1], [1841, 8, 1840, 9], [1, 4, 4, 17], [7, 6, 1842, 1843], [232, 1683, 1573, 1178], [1231, 396, 1738, 1825], [232, 1331, 1705, 1398], [628, 1177, 671, 342], [1175, 0, 0, 1847]] GL(2,Integers(1848)).subgroup(gens)
 
Copy content magma:Gens := [[1, 0, 8, 1], [1, 8, 0, 1], [1841, 8, 1840, 9], [1, 4, 4, 17], [7, 6, 1842, 1843], [232, 1683, 1573, 1178], [1231, 396, 1738, 1825], [232, 1331, 1705, 1398], [628, 1177, 671, 342], [1175, 0, 0, 1847]]; sub<GL(2,Integers(1848))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \), index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1841 & 8 \\ 1840 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1842 & 1843 \end{array}\right),\left(\begin{array}{rr} 232 & 1683 \\ 1573 & 1178 \end{array}\right),\left(\begin{array}{rr} 1231 & 396 \\ 1738 & 1825 \end{array}\right),\left(\begin{array}{rr} 232 & 1331 \\ 1705 & 1398 \end{array}\right),\left(\begin{array}{rr} 628 & 1177 \\ 671 & 342 \end{array}\right),\left(\begin{array}{rr} 1175 & 0 \\ 0 & 1847 \end{array}\right)$.

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

The torsion field $K:=\Q(E[1848])$ is a degree-$40874803200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1848\Z)$.

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
$2$ additive $4$ \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
$3$ nonsplit multiplicative $4$ \( 135520 = 2^{5} \cdot 5 \cdot 7 \cdot 11^{2} \)
$5$ split multiplicative $6$ \( 81312 = 2^{5} \cdot 3 \cdot 7 \cdot 11^{2} \)
$7$ nonsplit multiplicative $8$ \( 19360 = 2^{5} \cdot 5 \cdot 11^{2} \)
$11$ additive $62$ \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)

Isogenies

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

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 406560ch consists of 4 curves linked by isogenies of degrees dividing 4.

Twists

The minimal quadratic twist of this elliptic curve is 3360q3, its twist by $-11$.

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.