Properties

Label 422370e4
Conductor $422370$
Discriminant $-2.649\times 10^{20}$
j-invariant \( \frac{157376536199}{7722894400} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3-x^2+365445x-778480475\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-x^2z+365445xz^2-778480475z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+5847117x-49816903282\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, -1, 0, 365445, -778480475])
 
gp: E = ellinit([1, -1, 0, 365445, -778480475])
 
magma: E := EllipticCurve([1, -1, 0, 365445, -778480475]);
 
oscar: E = EllipticCurve([1, -1, 0, 365445, -778480475])
 
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

$P$ =  \(\left(21590, 3162605\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $7.0623981856651307612971397613$

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

Torsion generators

\( \left(\frac{3155}{4}, -\frac{3155}{8}\right) \) 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

\( \left(21590, 3162605\right) \), \( \left(21590, -3184195\right) \) Copy content Toggle raw display

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

Invariants

Conductor: \( 422370 \)  =  $2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 19^{2}$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-264867840399197505600 $  =  $-1 \cdot 2^{6} \cdot 3^{6} \cdot 5^{2} \cdot 13^{6} \cdot 19^{6} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{157376536199}{7722894400} \)  =  $2^{-6} \cdot 5^{-2} \cdot 13^{-6} \cdot 5399^{3}$
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: $2.5985486266749113160005888398\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $0.57702299275763624029845250539\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 

BSD invariants

Analytic rank: $1$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $7.0623981856651307612971397613\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.083538895936749603278080625225\dots$
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: $ 32 $  = $ 2\cdot2\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: $2$
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) $ ≈ $ 4.7198795767693484995433556557 $
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

$\displaystyle 4.719879577 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.083539 \cdot 7.062398 \cdot 32}{2^2} \approx 4.719879577$

# 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 422370.2.a.e

\( q - q^{2} + q^{4} - q^{5} - 4 q^{7} - q^{8} + q^{10} + 6 q^{11} - q^{13} + 4 q^{14} + q^{16} + 6 q^{17} + O(q^{20}) \) 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: 24634368
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: no
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$(j)_{-}$
$2$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$3$ $2$ $I_0^{*}$ Additive -1 2 6 0
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$13$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$19$ $2$ $I_0^{*}$ 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 4.6.0.5
$3$ 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 = [[13, 24, 28932, 28333], [25917, 21850, 6764, 11021], [14039, 0, 0, 29639], [14821, 10944, 0, 1], [19759, 18696, 4940, 29639], [1, 24, 0, 1], [29617, 24, 29616, 25], [15, 16, 314, 335], [18241, 24966, 2052, 3193], [1, 0, 24, 1], [22231, 10944, 11115, 1]]
 
GL(2,Integers(29640)).subgroup(gens)
 
Gens := [[13, 24, 28932, 28333], [25917, 21850, 6764, 11021], [14039, 0, 0, 29639], [14821, 10944, 0, 1], [19759, 18696, 4940, 29639], [1, 24, 0, 1], [29617, 24, 29616, 25], [15, 16, 314, 335], [18241, 24966, 2052, 3193], [1, 0, 24, 1], [22231, 10944, 11115, 1]];
 
sub<GL(2,Integers(29640))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 29640 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 19 \), index $384$, genus $9$, and generators

$\left(\begin{array}{rr} 13 & 24 \\ 28932 & 28333 \end{array}\right),\left(\begin{array}{rr} 25917 & 21850 \\ 6764 & 11021 \end{array}\right),\left(\begin{array}{rr} 14039 & 0 \\ 0 & 29639 \end{array}\right),\left(\begin{array}{rr} 14821 & 10944 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19759 & 18696 \\ 4940 & 29639 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 29617 & 24 \\ 29616 & 25 \end{array}\right),\left(\begin{array}{rr} 15 & 16 \\ 314 & 335 \end{array}\right),\left(\begin{array}{rr} 18241 & 24966 \\ 2052 & 3193 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 22231 & 10944 \\ 11115 & 1 \end{array}\right)$.

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

The torsion field $K:=\Q(E[29640])$ is a degree-$297375340953600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/29640\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 422370e consists of 4 curves linked by isogenies of degrees dividing 6.

Twists

The minimal quadratic twist of this elliptic curve is 130a4, its twist by $57$.

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.