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Properties

Label 405042.dp3
Conductor $405042$
Discriminant $3.253\times 10^{20}$
j-invariant \( \frac{62768149033310713}{6915442583808} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3-2988907x+1789355057\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-2988907xz^2+1789355057z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-3873623499x+83495770409862\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 0, 0, -2988907, 1789355057])
 
gp: E = ellinit([1, 0, 0, -2988907, 1789355057])
 
magma: E := EllipticCurve([1, 0, 0, -2988907, 1789355057]);
 
oscar: E = EllipticCurve([1, 0, 0, -2988907, 1789355057])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

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

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generator and height

$P$ =  \(\left(668, 9203\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $1.2021213845420618629433354202$

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

Torsion generators

\( \left(1246, -623\right) \) 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

\( \left(-1642, 48473\right) \), \( \left(-1642, -46831\right) \), \( \left(668, 9203\right) \), \( \left(668, -9871\right) \), \( \left(1246, -623\right) \), \( \left(6146, 461237\right) \), \( \left(6146, -467383\right) \) Copy content Toggle raw display

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

Invariants

Conductor: \( 405042 \)  =  $2 \cdot 3 \cdot 11 \cdot 17 \cdot 19^{2}$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $325343088860163694848 $  =  $2^{8} \cdot 3^{5} \cdot 11^{3} \cdot 17^{4} \cdot 19^{6} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{62768149033310713}{6915442583808} \)  =  $2^{-8} \cdot 3^{-5} \cdot 11^{-3} \cdot 17^{-4} \cdot 23^{3} \cdot 37^{3} \cdot 467^{3}$
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: $2.6693382981639647404933247507\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $1.1971188085807445104888110348\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $0.9879715979533532\dots$
Szpiro ratio: $4.363844970300918\dots$

BSD invariants

Analytic rank: $1$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $1.2021213845420618629433354202\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.16610611817210741775264573361\dots$
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: $ 480 $  = $ 2^{3}\cdot5\cdot3\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: $2$
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) $ ≈ $ 23.961566010955333330576203247 $
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

$\displaystyle 23.961566011 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.166106 \cdot 1.202121 \cdot 480}{2^2} \approx 23.961566011$

# 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 405042.2.a.dp

\( q + q^{2} + q^{3} + q^{4} + 2 q^{5} + q^{6} + 4 q^{7} + q^{8} + q^{9} + 2 q^{10} + q^{11} + q^{12} - 6 q^{13} + 4 q^{14} + 2 q^{15} + q^{16} - q^{17} + q^{18} + O(q^{20}) \) 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: 26542080
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

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$(j)_{-}$
$2$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$3$ $5$ $I_{5}$ Split multiplicative -1 1 5 5
$11$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$17$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$19$ $2$ $I_0^{*}$ Additive -1 2 6 0

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
$2$ 2B 4.6.0.1

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 = [[71251, 71250, 38722, 571], [1, 0, 8, 1], [62831, 0, 0, 85271], [1, 8, 0, 1], [10033, 76304, 35644, 49401], [1, 4, 4, 17], [85265, 8, 85264, 9], [7297, 7296, 81358, 43207], [74804, 62833, 23959, 58350], [7, 6, 85266, 85267], [21224, 17955, 73853, 62834]]
 
GL(2,Integers(85272)).subgroup(gens)
 
Gens := [[71251, 71250, 38722, 571], [1, 0, 8, 1], [62831, 0, 0, 85271], [1, 8, 0, 1], [10033, 76304, 35644, 49401], [1, 4, 4, 17], [85265, 8, 85264, 9], [7297, 7296, 81358, 43207], [74804, 62833, 23959, 58350], [7, 6, 85266, 85267], [21224, 17955, 73853, 62834]];
 
sub<GL(2,Integers(85272))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 85272 = 2^{3} \cdot 3 \cdot 11 \cdot 17 \cdot 19 \), index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 71251 & 71250 \\ 38722 & 571 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 62831 & 0 \\ 0 & 85271 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10033 & 76304 \\ 35644 & 49401 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 85265 & 8 \\ 85264 & 9 \end{array}\right),\left(\begin{array}{rr} 7297 & 7296 \\ 81358 & 43207 \end{array}\right),\left(\begin{array}{rr} 74804 & 62833 \\ 23959 & 58350 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 85266 & 85267 \end{array}\right),\left(\begin{array}{rr} 21224 & 17955 \\ 73853 & 62834 \end{array}\right)$.

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

The torsion field $K:=\Q(E[85272])$ is a degree-$195548795633664000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/85272\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 405042.dp consists of 4 curves linked by isogenies of degrees dividing 4.

Twists

The minimal quadratic twist of this elliptic curve is 1122.c3, its twist by $-19$.

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.