Properties

Label 2304c1
Conductor $2304$
Discriminant $373248$
j-invariant \( 8000 \)
CM yes (\(D=-8\))
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -30, -56])
 
gp: E = ellinit([0, 0, 0, -30, -56])
 
magma: E := EllipticCurve([0, 0, 0, -30, -56]);
 

\(y^2=x^3-30x-56\)  Toggle raw display

Mordell-Weil group structure

$\Z/{2}\Z$

Torsion generators

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

\( \left(-4, 0\right) \)  Toggle raw display

Integral points

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

\( \left(-4, 0\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 2304 \)  =  $2^{8} \cdot 3^{2}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $373248 $  =  $2^{9} \cdot 3^{6} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( 8000 \)  =  $2^{6} \cdot 5^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z[\sqrt{-2}]\) (potential complex multiplication)
Sato-Tate group: $N(\mathrm{U}(1))$
Faltings height: $-0.20124009630581192876727104788\dots$
Stable Faltings height: $-1.2704066260598257565278177574\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $0$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $2.0566953213264561394655623131\dots$
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: $ 4 $  = $ 2\cdot2 $
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) $ ≈ $ 2.0566953213264561394655623130780611252 $

Modular invariants

Modular form   2304.2.a.i

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);
 

\( q + 6q^{11} + 6q^{17} - 2q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 192
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 2 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$(j)_{-}$
$2$ $2$ $III$ Additive 1 8 9 0
$3$ $2$ $I_0^{*}$ Additive -1 2 6 0

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 $\ell$.

$p$-adic regulators

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

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

Iwasawa invariants

$p$ 2 3
Reduction type add add
$\lambda$-invariant(s) - -
$\mu$-invariant(s) - -

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

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 2304c consists of 2 curves linked by isogenies of degree 2.

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
$2$ \(\Q(\sqrt{2}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$4$ 4.0.18432.2 \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ 4.2.18432.2 \(\Z/6\Z\) Not in database
$4$ 4.0.6144.1 \(\Z/6\Z\) Not in database
$8$ 8.4.5435817984.2 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.339738624.10 \(\Z/3\Z \times \Z/6\Z\) Not in database
$8$ 8.4.1358954496.3 \(\Z/2\Z \times \Z/6\Z\) Not in database
$8$ 8.0.150994944.2 \(\Z/2\Z \times \Z/6\Z\) Not in database
$12$ 12.0.1521681143169024.7 \(\Z/18\Z\) Not in database
$16$ 16.0.118192468620711297024.15 \(\Z/4\Z \times \Z/4\Z\) Not in database
$16$ 16.0.29548117155177824256.4 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ 16.0.1846757322198614016.7 \(\Z/6\Z \times \Z/6\Z\) Not in database
$16$ 16.0.29548117155177824256.5 \(\Z/2\Z \times \Z/12\Z\) Not in database
$16$ 16.0.29548117155177824256.6 \(\Z/2\Z \times \Z/12\Z\) Not in database
$20$ 20.0.5016449852600330836716407488512.2 \(\Z/22\Z\) Not in database

We only show fields where the torsion growth is primitive.