# Properties

 Label 1850.h1 Conductor $1850$ Discriminant $-287100108800$ j-invariant $$-\frac{132384574175625}{11484004352}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3-x^2-3105x+72177$$ y^2+xy+y=x^3-x^2-3105x+72177 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-x^2z-3105xz^2+72177z^3$$ y^2z+xyz+yz^2=x^3-x^2z-3105xz^2+72177z^3 (dehomogenize, simplify) $$y^2=x^3-49675x+4569670$$ y^2=x^3-49675x+4569670 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, -1, 1, -3105, 72177])

gp: E = ellinit([1, -1, 1, -3105, 72177])

magma: E := EllipticCurve([1, -1, 1, -3105, 72177]);

oscar: E = EllipticCurve([1, -1, 1, -3105, 72177])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z$$

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generator and height

 $P$ = $$\left(183, 2276\right)$$ (183, 2276) $\hat{h}(P)$ ≈ $0.062324207193329351339671299421$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Integral points

$$\left(-57, 276\right)$$, $$\left(-57, -220\right)$$, $$\left(23, 100\right)$$, $$\left(23, -124\right)$$, $$\left(35, 56\right)$$, $$\left(35, -92\right)$$, $$\left(55, 228\right)$$, $$\left(55, -284\right)$$, $$\left(183, 2276\right)$$, $$\left(183, -2460\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$1850$$ = $2 \cdot 5^{2} \cdot 37$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-287100108800$ = $-1 \cdot 2^{23} \cdot 5^{2} \cdot 37^{2}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-\frac{132384574175625}{11484004352}$$ = $-1 \cdot 2^{-23} \cdot 3^{3} \cdot 5^{4} \cdot 37^{-2} \cdot 1987^{3}$ 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: $0.94288334637591242830309097047\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $0.67464369430356236586963108160\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $0.062324207193329351339671299421\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.95352589124303854065543625012\dots$ 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: $46$  = $23\cdot1\cdot2$ 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: $1$ 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: $L'(E,1)$ ≈ $2.7336762796616175143937526591$ 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

$\displaystyle 2.733676280 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.953526 \cdot 0.062324 \cdot 46}{1^2} \approx 2.733676280$

# 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("analytic 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

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

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);

Modular degree: 4416
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: yes
Manin constant: 1
comment: Manin constant

magma: ManinConstant(E);

## Local data

This elliptic curve is not semistable. There are 3 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $23$ $I_{23}$ Split multiplicative -1 1 23 23
$5$ $1$ $II$ Additive 1 2 2 0
$37$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

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
$2$ 2G 8.2.0.1

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 = [[5, 2, 5, 3], [1, 2, 0, 1], [7, 2, 6, 3], [1, 0, 2, 1], [1, 1, 7, 0], [7, 2, 7, 3]]

GL(2,Integers(8)).subgroup(gens)

Gens := [[5, 2, 5, 3], [1, 2, 0, 1], [7, 2, 6, 3], [1, 0, 2, 1], [1, 1, 7, 0], [7, 2, 7, 3]];

sub<GL(2,Integers(8))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 8.2.0.a.1, level $$8 = 2^{3}$$, index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 5 & 2 \\ 5 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 2 \\ 6 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 7 & 0 \end{array}\right),\left(\begin{array}{rr} 7 & 2 \\ 7 & 3 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[8])$ is a degree-$768$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has no rational isogenies. Its isogeny class 1850.h consists of this curve only.

## Twists

This elliptic curve is its own minimal quadratic twist.

## 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 $K$ $3$ 3.1.200.1 $$\Z/2\Z$$ Not in database $6$ 6.0.320000.1 $$\Z/2\Z \oplus \Z/2\Z$$ 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.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 split ss add ss ord ord ord ord ord ss ord split ord ord ord 3 1,1 - 1,1 1 3 1 1 1 1,1 1 2 1 1 1 0 0,0 - 0,0 0 0 0 0 0 0,0 0 0 0 0 0

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

## $p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.