# Properties

 Label 441f1 Conductor $441$ Discriminant $-107163$ j-invariant $$-\frac{28672}{3}$$ CM no Rank $1$ Torsion structure trivial

# Learn more

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+y=x^3-21x+40$$ y^2+y=x^3-21x+40 (homogenize, simplify) $$y^2z+yz^2=x^3-21xz^2+40z^3$$ y^2z+yz^2=x^3-21xz^2+40z^3 (dehomogenize, simplify) $$y^2=x^3-336x+2576$$ y^2=x^3-336x+2576 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 0, 1, -21, 40])

gp: E = ellinit([0, 0, 1, -21, 40])

magma: E := EllipticCurve([0, 0, 1, -21, 40]);

oscar: E = EllipticCurve([0, 0, 1, -21, 40])

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

 $P$ = $$\left(4, 4\right)$$ (4, 4) $\hat{h}(P)$ ≈ $0.076270034394571012613674763318$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Integral points

$$\left(-5, 4\right)$$, $$\left(-5, -5\right)$$, $$\left(1, 4\right)$$, $$\left(1, -5\right)$$, $$\left(2, 2\right)$$, $$\left(2, -3\right)$$, $$\left(4, 4\right)$$, $$\left(4, -5\right)$$, $$\left(46, 310\right)$$, $$\left(46, -311\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$441$$ = $3^{2} \cdot 7^{2}$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-107163$ = $-1 \cdot 3^{7} \cdot 7^{2}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-\frac{28672}{3}$$ = $-1 \cdot 2^{12} \cdot 3^{-1} \cdot 7$ 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: $-0.29578137208499745632736253347\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-1.1694058745949378528758772758\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $0.9123874511245065\dots$ Szpiro ratio: $3.434588835051495\dots$

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $0.076270034394571012613674763318\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $3.2597903746182327614046723214\dots$ 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: $4$  = $2^{2}\cdot1$ 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: $L'(E,1)$ ≈ $0.99449729596489655501469818657$ 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

$\displaystyle 0.994497296 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 3.259790 \cdot 0.076270 \cdot 4}{1^2} \approx 0.994497296$

# 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

$$q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{10} + 2 q^{11} - q^{13} - 4 q^{16} - q^{19} + O(q^{20})$$

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);

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

Modular degree: 48
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 2 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$3$ $4$ $I_{1}^{*}$ Additive -1 2 7 1
$7$ $1$ $II$ Additive -1 2 2 0

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
$13$ 13B.4.1 13.28.0.1

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 = [[335, 26, 78, 251], [14, 23, 325, 339], [1, 26, 0, 379], [181, 520, 169, 207], [1, 0, 26, 1], [1, 26, 0, 1], [521, 26, 520, 27]]

GL(2,Integers(546)).subgroup(gens)

Gens := [[335, 26, 78, 251], [14, 23, 325, 339], [1, 26, 0, 379], [181, 520, 169, 207], [1, 0, 26, 1], [1, 26, 0, 1], [521, 26, 520, 27]];

sub<GL(2,Integers(546))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$, index $336$, genus $9$, and generators

$\left(\begin{array}{rr} 335 & 26 \\ 78 & 251 \end{array}\right),\left(\begin{array}{rr} 14 & 23 \\ 325 & 339 \end{array}\right),\left(\begin{array}{rr} 1 & 26 \\ 0 & 379 \end{array}\right),\left(\begin{array}{rr} 181 & 520 \\ 169 & 207 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 26 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 26 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 521 & 26 \\ 520 & 27 \end{array}\right)$.

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

The torsion field $K:=\Q(E[546])$ is a degree-$45287424$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/546\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 13.
Its isogeny class 441f consists of 2 curves linked by isogenies of degree 13.

## Twists

The minimal quadratic twist of this elliptic curve is 147c1, its twist by $-3$.

## 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 $K$ $3$ 3.1.588.1 $$\Z/2\Z$$ Not in database $6$ 6.0.1037232.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $6$ $$\Q(\zeta_{21})^+$$ $$\Z/13\Z$$ Not in database $8$ 8.2.20841167403.2 $$\Z/3\Z$$ Not in database $12$ deg 12 $$\Z/4\Z$$ Not in database $18$ 18.6.382755853108052250624.1 $$\Z/26\Z$$ 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 ss add ord add ord ord ss ord ss ord ord ord ord ord ord ? - 1 - 1 1 1,1 1 1,1 1 1 1 1 1 1 ? - 0 - 0 0 0,0 0 0,0 0 0 0 0 0 0

An entry ? indicates that the invariants have not yet been computed.

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.