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Properties

Label 6240.e4
Conductor $6240$
Discriminant $-219348480$
j-invariant \( \frac{240641848}{428415} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-x^2+104x-620\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-x^2z+104xz^2-620z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+8397x-426762\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, -1, 0, 104, -620])
 
gp: E = ellinit([0, -1, 0, 104, -620])
 
magma: E := EllipticCurve([0, -1, 0, 104, -620]);
 
oscar: E = EllipticCurve([0, -1, 0, 104, -620])
 
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

\( \left(5, 0\right) \) 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

\( \left(5, 0\right) \) Copy content Toggle raw display

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

Invariants

Conductor: \( 6240 \)  =  $2^{5} \cdot 3 \cdot 5 \cdot 13$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-219348480 $  =  $-1 \cdot 2^{9} \cdot 3 \cdot 5 \cdot 13^{4} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{240641848}{428415} \)  =  $2^{3} \cdot 3^{-1} \cdot 5^{-1} \cdot 13^{-4} \cdot 311^{3}$
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: $0.28533801215580929454920833919\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $-0.23452237326414968751371575190\dots$
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: $0.92938816891485510908817725344\dots$
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: $ 8 $  = $ 2\cdot1\cdot1\cdot2^{2} $
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: $1$ (exact)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Special value: $ L(E,1) $ ≈ $ 1.8587763378297102181763545069 $
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

$\displaystyle 1.858776338 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.929388 \cdot 1.000000 \cdot 8}{2^2} \approx 1.858776338$

# 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("anayltic 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   6240.2.a.e

\( q - q^{3} - q^{5} + q^{9} + 4 q^{11} + q^{13} + q^{15} + 6 q^{17} + 4 q^{19} + O(q^{20}) \) 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: 2048
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 4 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $I_0^{*}$ Additive 1 5 9 0
$3$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$5$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$13$ $4$ $I_{4}$ Split 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 $\ell$ 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 = [[1081, 8, 1204, 33], [628, 1, 1271, 6], [1, 0, 8, 1], [1, 8, 0, 1], [1, 4, 4, 17], [7, 6, 1554, 1555], [1553, 8, 1552, 9], [1369, 1368, 598, 1375], [1048, 3, 1045, 2], [1368, 593, 187, 174]]
 
GL(2,Integers(1560)).subgroup(gens)
 
Gens := [[1081, 8, 1204, 33], [628, 1, 1271, 6], [1, 0, 8, 1], [1, 8, 0, 1], [1, 4, 4, 17], [7, 6, 1554, 1555], [1553, 8, 1552, 9], [1369, 1368, 598, 1375], [1048, 3, 1045, 2], [1368, 593, 187, 174]];
 
sub<GL(2,Integers(1560))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 1081 & 8 \\ 1204 & 33 \end{array}\right),\left(\begin{array}{rr} 628 & 1 \\ 1271 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1554 & 1555 \end{array}\right),\left(\begin{array}{rr} 1553 & 8 \\ 1552 & 9 \end{array}\right),\left(\begin{array}{rr} 1369 & 1368 \\ 598 & 1375 \end{array}\right),\left(\begin{array}{rr} 1048 & 3 \\ 1045 & 2 \end{array}\right),\left(\begin{array}{rr} 1368 & 593 \\ 187 & 174 \end{array}\right)$.

The torsion field $K:=\Q(E[1560])$ is a degree-$2396172$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.

$p$-adic regulators

All $p$-adic regulators are identically $1$ since the rank is $0$.

Iwasawa invariants

$p$ 2 3 5 13
Reduction type add nonsplit nonsplit split
$\lambda$-invariant(s) - 0 0 1
$\mu$-invariant(s) - 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 6240.e consists of 4 curves linked by isogenies of degrees dividing 4.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-30}) \) \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{10}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{-3}) \) \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-3}, \sqrt{10})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ 8.0.191102976000000.15 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ deg 8 \(\Z/8\Z\) Not in database
$8$ 8.0.136452302438400.3 \(\Z/8\Z\) Not in database
$8$ deg 8 \(\Z/6\Z\) Not in database
$16$ deg 16 \(\Z/4\Z \oplus \Z/4\Z\) Not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$16$ deg 16 \(\Z/12\Z\) Not in database
$16$ deg 16 \(\Z/12\Z\) Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.