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Properties

Label 445280bq1
Conductor $445280$
Discriminant $-1.229\times 10^{18}$
j-invariant \( -\frac{319387394710714176}{1399205} \)
CM no
Rank $1$
Torsion structure trivial

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Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-136363612x-612908803936\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-136363612xz^2-612908803936z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-136363612x-612908803936\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, 0, 0, -136363612, -612908803936])
 
gp: E = ellinit([0, 0, 0, -136363612, -612908803936])
 
magma: E := EllipticCurve([0, 0, 0, -136363612, -612908803936]);
 
oscar: E = elliptic_curve([0, 0, 0, -136363612, -612908803936])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z\)

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generator and height

$P$ =  \(\left(\frac{17242612928136049509285165067607557642954563570005542824901366747038225040622557769407339726744861267008940}{907266733127584089284155360225083077903106187106927273986415689476657396262727285305922562080805088209}, \frac{1653250434995854700072201316099462934979786736192559296955879539940213818576387465518817914848166884117377481213005535035060194118023225682992708099406698958124}{864176555936801811780606623036388298896122125836828496949553165551307412565834450037141899297577191405956480220152164480981012241747240167123286990833177}\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $241.09379298842602217720428829$

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

Integral points

None

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

Invariants

Conductor: \( 445280 \)  =  $2^{5} \cdot 5 \cdot 11^{2} \cdot 23$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-1228521546095022080 $  =  $-1 \cdot 2^{12} \cdot 5 \cdot 11^{8} \cdot 23^{4} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( -\frac{319387394710714176}{1399205} \)  =  $-1 \cdot 2^{6} \cdot 3^{3} \cdot 5^{-1} \cdot 7^{3} \cdot 11 \cdot 23^{-4} \cdot 3659^{3}$
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: $3.0995739879790099128858899375\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $0.80782995888681757409402876407\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $0.990378361798107\dots$
Szpiro ratio: $5.213264557826692\dots$

BSD invariants

Analytic rank: $1$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $241.09379298842602217720428829\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.022089960272766709522953921225\dots$
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: $ 4 $  = $ 2\cdot1\cdot1\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: $1$
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) $ ≈ $ 21.303009236499887570858964903 $
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

$\displaystyle 21.303009236 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.022090 \cdot 241.093793 \cdot 4}{1^2} \approx 21.303009236$

# 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 analytic 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 445280.2.a.bq

\( q + 3 q^{3} + q^{5} + q^{7} + 6 q^{9} + 4 q^{13} + 3 q^{15} + O(q^{20}) \) 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: 57987072
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 4 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $v_p(N)$ $v_p(\Delta)$ $v_p(\mathrm{den}(j))$
$2$ $2$ $I_{3}^{*}$ additive 1 5 12 0
$5$ $1$ $I_{1}$ split multiplicative -1 1 1 1
$11$ $1$ $IV^{*}$ additive -1 2 8 0
$23$ $2$ $I_{4}$ nonsplit multiplicative 1 1 4 4

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 $\ell$.

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 = [[1, 2, 0, 1], [1, 1, 19, 0], [17, 2, 17, 3], [1, 0, 2, 1], [11, 2, 11, 3], [19, 2, 18, 3]]
 
GL(2,Integers(20)).subgroup(gens)
 
Gens := [[1, 2, 0, 1], [1, 1, 19, 0], [17, 2, 17, 3], [1, 0, 2, 1], [11, 2, 11, 3], [19, 2, 18, 3]];
 
sub<GL(2,Integers(20))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 20.2.0.a.1, level \( 20 = 2^{2} \cdot 5 \), index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 19 & 0 \end{array}\right),\left(\begin{array}{rr} 17 & 2 \\ 17 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 11 & 3 \end{array}\right),\left(\begin{array}{rr} 19 & 2 \\ 18 & 3 \end{array}\right)$.

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

The torsion field $K:=\Q(E[20])$ is a degree-$23040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/20\Z)$.

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 $2$ \( 605 = 5 \cdot 11^{2} \)
$5$ split multiplicative $6$ \( 89056 = 2^{5} \cdot 11^{2} \cdot 23 \)
$11$ additive $52$ \( 3680 = 2^{5} \cdot 5 \cdot 23 \)
$23$ nonsplit multiplicative $24$ \( 19360 = 2^{5} \cdot 5 \cdot 11^{2} \)

Isogenies

gp: ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 445280bq consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 445280e1, its twist by $44$.

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.