# Properties

 Label 308550h4 Conductor $308550$ Discriminant $3.988\times 10^{28}$ j-invariant $$\frac{1696892787277117093383481}{1440538624914939000}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -7516845250, 250655549321500])

gp: E = ellinit([1, 1, 0, -7516845250, 250655549321500])

magma: E := EllipticCurve([1, 1, 0, -7516845250, 250655549321500]);

$$y^2+xy=x^3+x^2-7516845250x+250655549321500$$

## Mordell-Weil group structure

$\Z/{2}\Z$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(\frac{204635}{4}, -\frac{204635}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$308550$$ = $2 \cdot 3 \cdot 5^{2} \cdot 11^{2} \cdot 17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $39875031982702097652796875000$ = $2^{3} \cdot 3^{3} \cdot 5^{9} \cdot 11^{18} \cdot 17$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{1696892787277117093383481}{1440538624914939000}$$ = $2^{-3} \cdot 3^{-3} \cdot 5^{-3} \cdot 11^{-12} \cdot 17^{-1} \cdot 181^{3} \cdot 227^{3} \cdot 2903^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $4.4151478971942426924481478121\dots$ Stable Faltings height: $2.4114813045780072331167963565\dots$

## 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: $0.036074346213706753174798604758\dots$ 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: $8$  = $1\cdot1\cdot2\cdot2^{2}\cdot1$ 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: $4$ = $2^2$ (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: $L(E,1)$ ≈ $0.28859476970965402539838883806$

## Modular invariants

Modular form 308550.2.a.h

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 637009920 $\Gamma_0(N)$-optimal: no 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$(j)_{-}$
$2$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$3$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$5$ $2$ $I_{3}^{*}$ Additive 1 2 9 3
$11$ $4$ $I_{12}^{*}$ Additive -1 2 18 12
$17$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

## 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
$3$ 3B 3.4.0.1

## $p$-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 non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3, 4, 6 and 12.
Its isogeny class 308550h consists of 8 curves linked by isogenies of degrees dividing 12.

## 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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{510})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{330})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{187})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-55})$$ $$\Z/6\Z$$ Not in database $4$ $$\Q(\sqrt{187}, \sqrt{330})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-55}, \sqrt{510})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database $4$ $$\Q(\sqrt{-6}, \sqrt{-55})$$ $$\Z/12\Z$$ Not in database $4$ $$\Q(\sqrt{-55}, \sqrt{-85})$$ $$\Z/12\Z$$ Not in database $6$ 6.2.375186772125.3 $$\Z/6\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \times \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/24\Z$$ Not in database $16$ Deg 16 $$\Z/24\Z$$ Not in database $18$ 18.0.183091994234716348674222198885513377504241943359375.1 $$\Z/18\Z$$ 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.