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Properties

Label 323400.cz4
Conductor $323400$
Discriminant $-1.085\times 10^{19}$
j-invariant \( \frac{20777545136}{23059575} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-x^2+445492x-109790988\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-x^2z+445492xz^2-109790988z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+36084825x-79929375750\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 0, 445492, -109790988])
 
gp: E = ellinit([0, -1, 0, 445492, -109790988])
 
magma: E := EllipticCurve([0, -1, 0, 445492, -109790988]);
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(761, 25872\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $2.8027776489549379934294592470$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(222, 0\right) \) Copy content Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(222, 0\right) \), \((761,\pm 25872)\), \((1322,\pm 52800)\) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 323400 \)  =  $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 11$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-10851743756700000000 $  =  $-1 \cdot 2^{8} \cdot 3^{2} \cdot 5^{8} \cdot 7^{7} \cdot 11^{4} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{20777545136}{23059575} \)  =  $2^{4} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-1} \cdot 11^{-4} \cdot 1091^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.3390760866545010248480230791\dots$
Stable Faltings height: $0.099303935536497312050145626457\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $2.8027776489549379934294592470\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.12286626949298812550700016981\dots$
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: $ 64 $  = $ 2\cdot2\cdot2\cdot2\cdot2^{2} $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $2$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Special value: $ L'(E,1) $ ≈ $ 5.5098693431267372760037761151 $

Modular invariants

Modular form 323400.2.a.cz

sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 
magma: ModularForm(E);
 

\( q - q^{3} + q^{9} + q^{11} - 2 q^{13} - 2 q^{17} + O(q^{20}) \) Copy content Toggle raw display

For more coefficients, see the Downloads section to the right.

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 4718592
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $I_{1}^{*}$ Additive 1 3 8 0
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$5$ $2$ $I_{2}^{*}$ Additive 1 2 8 2
$7$ $2$ $I_{1}^{*}$ Additive -1 2 7 1
$11$ $4$ $I_{4}$ Split multiplicative -1 1 4 4

Galois representations

sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

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

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 323400.cz 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
$2$ \(\Q(\sqrt{-7}) \) \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{5}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{-35}) \) \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{5}, \sqrt{-7})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ 8.0.1524731040000.52 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/8\Z\) Not in database
$8$ 8.0.1439868559360000.126 \(\Z/8\Z\) 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.