# Properties

 Label 10944bt1 Conductor $10944$ Discriminant $-52344812736$ j-invariant $$\frac{841232384}{1121931}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2=x^3+708x-8282$$ y^2=x^3+708x-8282 (homogenize, simplify) $$y^2z=x^3+708xz^2-8282z^3$$ y^2z=x^3+708xz^2-8282z^3 (dehomogenize, simplify) $$y^2=x^3+708x-8282$$ y^2=x^3+708x-8282 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 0, 0, 708, -8282])

gp: E = ellinit([0, 0, 0, 708, -8282])

magma: E := EllipticCurve([0, 0, 0, 708, -8282]);

oscar: E = EllipticCurve([0, 0, 0, 708, -8282])

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

 $P$ = $$\left(143, 1737\right)$$ (143, 1737) $\hat{h}(P)$ ≈ $3.9606265424671759369931658791$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Integral points

$$(143,\pm 1737)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$10944$$ = $2^{6} \cdot 3^{2} \cdot 19$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-52344812736$ = $-1 \cdot 2^{6} \cdot 3^{16} \cdot 19$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{841232384}{1121931}$$ = $2^{12} \cdot 3^{-10} \cdot 19^{-1} \cdot 59^{3}$ 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: $0.74256802962540580066960244292\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.15331170498862169973663623627\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.0048983874206856\dots$ Szpiro ratio: $3.394082766116465\dots$

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $3.9606265424671759369931658791\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.59860384533613063924258124751\dots$ 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: $2$  = $1\cdot2\cdot1$ 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: $1$ ( rounded) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L'(E,1)$ ≈ $4.7416925565223904673924260916$ 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

$\displaystyle 4.741692557 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.598604 \cdot 3.960627 \cdot 2}{1^2} \approx 4.741692557$

# 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

$$q + q^{5} - 3 q^{7} + 3 q^{11} + 6 q^{13} - 3 q^{17} - q^{19} + O(q^{20})$$

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: 7680
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$(j)_{-}$
$2$ $1$ $II$ Additive -1 6 6 0
$3$ $2$ $I_{10}^{*}$ Additive -1 2 16 10
$19$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

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
$5$ 5B.4.1 5.12.0.1

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 = [[6, 13, 2225, 2161], [2099, 750, 2100, 749], [2279, 750, 0, 683], [886, 765, 1605, 946], [1, 0, 10, 1], [1139, 0, 0, 2279], [2271, 10, 2270, 11], [1514, 747, 2145, 689], [1519, 0, 0, 2279], [569, 0, 0, 2279], [1, 10, 0, 1]]

GL(2,Integers(2280)).subgroup(gens)

Gens := [[6, 13, 2225, 2161], [2099, 750, 2100, 749], [2279, 750, 0, 683], [886, 765, 1605, 946], [1, 0, 10, 1], [1139, 0, 0, 2279], [2271, 10, 2270, 11], [1514, 747, 2145, 689], [1519, 0, 0, 2279], [569, 0, 0, 2279], [1, 10, 0, 1]];

sub<GL(2,Integers(2280))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19$$, index $48$, genus $1$, and generators

$\left(\begin{array}{rr} 6 & 13 \\ 2225 & 2161 \end{array}\right),\left(\begin{array}{rr} 2099 & 750 \\ 2100 & 749 \end{array}\right),\left(\begin{array}{rr} 2279 & 750 \\ 0 & 683 \end{array}\right),\left(\begin{array}{rr} 886 & 765 \\ 1605 & 946 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1139 & 0 \\ 0 & 2279 \end{array}\right),\left(\begin{array}{rr} 2271 & 10 \\ 2270 & 11 \end{array}\right),\left(\begin{array}{rr} 1514 & 747 \\ 2145 & 689 \end{array}\right),\left(\begin{array}{rr} 1519 & 0 \\ 0 & 2279 \end{array}\right),\left(\begin{array}{rr} 569 & 0 \\ 0 & 2279 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right)$.

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

The torsion field $K:=\Q(E[2280])$ is a degree-$90773913600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2280\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 5.
Its isogeny class 10944bt consists of 2 curves linked by isogenies of degree 5.

## Twists

The minimal quadratic twist of this elliptic curve is 57c1, its twist by $24$.

## 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 $K$ $2$ $$\Q(\sqrt{6})$$ $$\Z/5\Z$$ Not in database $3$ 3.1.76.1 $$\Z/2\Z$$ Not in database $6$ 6.0.109744.2 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $6$ 6.2.19961856.2 $$\Z/10\Z$$ Not in database $8$ deg 8 $$\Z/3\Z$$ Not in database $12$ deg 12 $$\Z/4\Z$$ Not in database $12$ deg 12 $$\Z/2\Z \oplus \Z/10\Z$$ Not in database $16$ deg 16 $$\Z/15\Z$$ Not in database $20$ 20.0.558110376910994221642395451392000000000000000.1 $$\Z/5\Z$$ 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 add add ord ord ord ord ord nonsplit ord ord ord ord ord ord ord - - 5 3 3 3 1 1 1 1 1 1 1 1 1 - - 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.