Properties

Label 407330.h2
Conductor $407330$
Discriminant $3.044\times 10^{25}$
j-invariant \( \frac{1098325674097093229481}{205612182617187500} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3-x^2-113708120x+383887936196\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-x^2z-113708120xz^2+383887936196z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-1819329923x+24567008586622\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, -1, 0, -113708120, 383887936196])
 
gp: E = ellinit([1, -1, 0, -113708120, 383887936196])
 
magma: E := EllipticCurve([1, -1, 0, -113708120, 383887936196]);
 
oscar: E = EllipticCurve([1, -1, 0, -113708120, 383887936196])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

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

magma: MordellWeilGroup(E);
 

Torsion generators

\( \left(-\frac{48253}{4}, \frac{48253}{8}\right) \) 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

None

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

Invariants

Conductor: \( 407330 \)  =  $2 \cdot 5 \cdot 7 \cdot 11 \cdot 23^{2}$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $30437982242965698242187500 $  =  $2^{2} \cdot 5^{20} \cdot 7^{2} \cdot 11 \cdot 23^{6} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{1098325674097093229481}{205612182617187500} \)  =  $2^{-2} \cdot 3^{3} \cdot 5^{-20} \cdot 7^{-2} \cdot 11^{-1} \cdot 37^{3} \cdot 92951^{3}$
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: $3.6077609544731212038721686498\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $2.0400138465085463584687922339\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 

BSD invariants

Analytic rank: $0$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $1$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.062785491579659027662248427011\dots$
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: $ 16 $  = $ 2\cdot2\cdot2\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: $2$
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: $4$ = $2^2$ ( exact)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Special value: $ L(E,1) $ ≈ $ 1.0045678652745444425959748322 $
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

$\displaystyle 1.004567865 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{4 \cdot 0.062785 \cdot 1.000000 \cdot 16}{2^2} \approx 1.004567865$

# 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 407330.2.a.h

\( q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} - 3 q^{9} + q^{10} + q^{11} + 2 q^{13} - q^{14} + q^{16} - 6 q^{17} + 3 q^{18} - 4 q^{19} + O(q^{20}) \) 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: 126156800
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: no
Manin constant: 1 (conditional*)
comment: Manin constant
 
magma: ManinConstant(E);
 
* The Manin constant is correct provided that curve 407330.h4 is optimal.

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$(j)_{-}$
$2$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$5$ $2$ $I_{20}$ Non-split multiplicative 1 1 20 20
$7$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$11$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$23$ $2$ $I_0^{*}$ 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 = [[10113, 8, 10112, 9], [1, 0, 8, 1], [3888, 9683, 9085, 6602], [1, 8, 0, 1], [1, 4, 4, 17], [7, 6, 10114, 10115], [8097, 2208, 8188, 8833], [6159, 0, 0, 10119], [8856, 6003, 4899, 7568], [8419, 8418, 1058, 10075]]
 
GL(2,Integers(10120)).subgroup(gens)
 
Gens := [[10113, 8, 10112, 9], [1, 0, 8, 1], [3888, 9683, 9085, 6602], [1, 8, 0, 1], [1, 4, 4, 17], [7, 6, 10114, 10115], [8097, 2208, 8188, 8833], [6159, 0, 0, 10119], [8856, 6003, 4899, 7568], [8419, 8418, 1058, 10075]];
 
sub<GL(2,Integers(10120))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10120 = 2^{3} \cdot 5 \cdot 11 \cdot 23 \), index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 10113 & 8 \\ 10112 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 3888 & 9683 \\ 9085 & 6602 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 10114 & 10115 \end{array}\right),\left(\begin{array}{rr} 8097 & 2208 \\ 8188 & 8833 \end{array}\right),\left(\begin{array}{rr} 6159 & 0 \\ 0 & 10119 \end{array}\right),\left(\begin{array}{rr} 8856 & 6003 \\ 4899 & 7568 \end{array}\right),\left(\begin{array}{rr} 8419 & 8418 \\ 1058 & 10075 \end{array}\right)$.

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

The torsion field $K:=\Q(E[10120])$ is a degree-$54168846336000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10120\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 407330.h consists of 4 curves linked by isogenies of degrees dividing 4.

Twists

The minimal quadratic twist of this elliptic curve is 770.c2, its twist by $-23$.

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 $0$.