# Properties

 Label 147c2 Conductor $147$ Discriminant $-78121827$ j-invariant $$-\frac{1713910976512}{1594323}$$ CM no Rank $0$ Torsion structure trivial

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

This isogeny class and its quadratic twist by $\Q(\sqrt{-3})$ are the ones of minimal conductor with a $13$-isogeny.

## Simplified equation

 $$y^2+y=x^3-x^2-912x+10919$$ y^2+y=x^3-x^2-912x+10919 (homogenize, simplify) $$y^2z+yz^2=x^3-x^2z-912xz^2+10919z^3$$ y^2z+yz^2=x^3-x^2z-912xz^2+10919z^3 (dehomogenize, simplify) $$y^2=x^3-1182384x+495261648$$ y^2=x^3-1182384x+495261648 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, -1, 1, -912, 10919])

gp: E = ellinit([0, -1, 1, -912, 10919])

magma: E := EllipticCurve([0, -1, 1, -912, 10919]);

oscar: E = EllipticCurve([0, -1, 1, -912, 10919])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

trivial

magma: MordellWeilGroup(E);

## Integral points

None

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$147$$ = $3 \cdot 7^{2}$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-78121827$ = $-1 \cdot 3^{13} \cdot 7^{2}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-\frac{1713910976512}{1594323}$$ = $-1 \cdot 2^{12} \cdot 3^{-13} \cdot 7 \cdot 17^{3} \cdot 23^{3}$ 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: $0.43738716231171606600175856885\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $0.11306880413583051515086644494\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## BSD invariants

 Analytic rank: $0$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $1.9200537453490419433285021703\dots$ 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: $1$ 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: $L(E,1)$ ≈ $1.9200537453490419433285021703$ 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

$\displaystyle 1.920053745 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.920054 \cdot 1.000000 \cdot 1}{1^2} \approx 1.920053745$

# 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

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

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: 78
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: no
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$(j)_{-}$
$3$ $1$ $I_{13}$ Non-split multiplicative 1 1 13 13
$7$ $1$ $II$ 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.4.2 13.28.0.2

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 = [[485, 26, 156, 101], [15, 26, 130, 287], [14, 23, 325, 339], [1, 0, 26, 1], [365, 26, 377, 339], [1, 26, 0, 1], [521, 26, 520, 27]]

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

Gens := [[485, 26, 156, 101], [15, 26, 130, 287], [14, 23, 325, 339], [1, 0, 26, 1], [365, 26, 377, 339], [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

$\left(\begin{array}{rr} 485 & 26 \\ 156 & 101 \end{array}\right),\left(\begin{array}{rr} 15 & 26 \\ 130 & 287 \end{array}\right),\left(\begin{array}{rr} 14 & 23 \\ 325 & 339 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 26 & 1 \end{array}\right),\left(\begin{array}{rr} 365 & 26 \\ 377 & 339 \end{array}\right),\left(\begin{array}{rr} 1 & 26 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 521 & 26 \\ 520 & 27 \end{array}\right)$.

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 147c consists of 2 curves linked by isogenies of degree 13.

## 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 $K$ $3$ 3.1.588.1 $$\Z/2\Z$$ Not in database $6$ 6.0.1037232.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $8$ 8.2.20841167403.1 $$\Z/3\Z$$ Not in database $12$ deg 12 $$\Z/4\Z$$ Not in database $12$ 12.12.506240953553539690213.1 $$\Z/13\Z$$ 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 nonsplit ord add ord ord ss ord ss ord ord ord ord ord ord 0,1 0 0 - 0 0 0,0 0 0,0 0 0 0 0 0 0 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

All $p$-adic regulators are identically $1$ since the rank is $0$.