# Properties

 Label 327990v1 Conductor $327990$ Discriminant $-2.808\times 10^{24}$ j-invariant $$-\frac{3198929765521}{6673387500}$$ CM no Rank $0$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -23001368, -91116025342])

gp: E = ellinit([1, 0, 1, -23001368, -91116025342])

magma: E := EllipticCurve([1, 0, 1, -23001368, -91116025342]);

$$y^2+xy+y=x^3-23001368x-91116025342$$

trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$327990$$ = $2 \cdot 3 \cdot 5 \cdot 13 \cdot 29^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-2807542391865145100887500$ = $-1 \cdot 2^{2} \cdot 3^{5} \cdot 5^{5} \cdot 13^{3} \cdot 29^{10}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{3198929765521}{6673387500}$$ = $-1 \cdot 2^{-2} \cdot 3^{-5} \cdot 5^{-5} \cdot 7^{3} \cdot 13^{-3} \cdot 29^{2} \cdot 223^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $3.3802944202801914173518562161\dots$ Stable Faltings height: $0.57421456195812972803246285580\dots$

## 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: $0.032336304246049022277161516108\dots$ 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: $150$  = $2\cdot5\cdot5\cdot3\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $1$ 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: $L(E,1)$ ≈ $4.8504456369073533415742274161931508896$

## Modular invariants

Modular form 327990.2.a.v

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);

$$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + 5q^{7} - q^{8} + q^{9} - q^{10} + q^{11} + q^{12} + q^{13} - 5q^{14} + q^{15} + q^{16} + 2q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 102312000 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$3$ $5$ $I_{5}$ Split multiplicative -1 1 5 5
$5$ $5$ $I_{5}$ Split multiplicative -1 1 5 5
$13$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$29$ $1$ $II^{*}$ Additive 1 2 10 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 $0$.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has no rational isogenies. Its isogeny class 327990v consists of this curve only.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.163995.1 $$\Z/2\Z$$ Not in database $6$ 6.0.5244400204875.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ Deg 8 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.