# Properties

 Label 266910.ck2 Conductor $266910$ Discriminant $7.571\times 10^{26}$ j-invariant $$\frac{1144081181272479239133643526834383042983361}{757051842880264964991840000}$$ CM no Rank $1$ Torsion structure $$\Z/{8}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy=x^3-2178935510140x+1237982811464867600$$ y^2+xy=x^3-2178935510140x+1237982811464867600 (homogenize, simplify) $$y^2z+xyz=x^3-2178935510140xz^2+1237982811464867600z^3$$ y^2z+xyz=x^3-2178935510140xz^2+1237982811464867600z^3 (dehomogenize, simplify) $$y^2=x^3-2823900421141467x+57759334523406126169974$$ y^2=x^3-2823900421141467x+57759334523406126169974 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 0, 0, -2178935510140, 1237982811464867600])

gp: E = ellinit([1, 0, 0, -2178935510140, 1237982811464867600])

magma: E := EllipticCurve([1, 0, 0, -2178935510140, 1237982811464867600]);

oscar: E = elliptic_curve([1, 0, 0, -2178935510140, 1237982811464867600])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z \oplus \Z/{8}\Z$$

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generator and height

 $P$ = $$\left(\frac{69048190}{81}, -\frac{68757760}{729}\right)$$ (69048190/81, -68757760/729) $\hat{h}(P)$ ≈ $11.256657955703678617848108104$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(852260, -392920\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(852260, -392920\right)$$, $$\left(852260, -459340\right)$$, $$\left(853880, 2197460\right)$$, $$\left(853880, -3051340\right)$$, $$\left(985100, 217398260\right)$$, $$\left(985100, -218383360\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$266910$$ = $2 \cdot 3 \cdot 5 \cdot 7 \cdot 31 \cdot 41$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $757051842880264964991840000$ = $2^{8} \cdot 3^{32} \cdot 5^{4} \cdot 7^{2} \cdot 31 \cdot 41^{2}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{1144081181272479239133643526834383042983361}{757051842880264964991840000}$$ = $2^{-8} \cdot 3^{-32} \cdot 5^{-4} \cdot 7^{-2} \cdot 31^{-1} \cdot 41^{-2} \cdot 47^{3} \cdot 63839^{3} \cdot 34857937^{3}$ 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: $5.3743425335345865269406351819\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $5.3743425335345865269406351819\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.0391198239885415\dots$ Szpiro ratio: $7.7507609681964835\dots$

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $11.256657955703678617848108104\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.021858830300185668783235423915\dots$ 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: $4096$  = $2^{3}\cdot2^{5}\cdot2^{2}\cdot2\cdot1\cdot2$ 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: $8$ 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: $L'(E,1)$ ≈ $15.747672064061544830669644969$ 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

$\displaystyle 15.747672064 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.021859 \cdot 11.256658 \cdot 4096}{8^2} \approx 15.747672064$

# 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

Modular form 266910.2.a.ck

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + q^{10} - 4 q^{11} + q^{12} - 2 q^{13} - q^{14} + q^{15} + q^{16} + 2 q^{17} + q^{18} + 4 q^{19} + O(q^{20})$$

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: 3321888768
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 semistable. There are 6 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $v_p(N)$ $v_p(\Delta)$ $v_p(\mathrm{den}(j))$
$2$ $8$ $I_{8}$ split multiplicative -1 1 8 8
$3$ $32$ $I_{32}$ split multiplicative -1 1 32 32
$5$ $4$ $I_{4}$ split multiplicative -1 1 4 4
$7$ $2$ $I_{2}$ nonsplit multiplicative 1 1 2 2
$31$ $1$ $I_{1}$ nonsplit multiplicative 1 1 1 1
$41$ $2$ $I_{2}$ 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
$2$ 2B 16.96.0.95

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 = [[1, 0, 32, 1], [1, 32, 0, 1], [203329, 32, 203328, 33], [5, 28, 68, 381], [32832, 29, 199627, 2562], [38162, 19, 87215, 202318], [23, 18, 200798, 201355], [177943, 2, 177914, 203343], [162711, 26, 38134, 200491], [39703, 26, 124774, 875]]

GL(2,Integers(203360)).subgroup(gens)

Gens := [[1, 0, 32, 1], [1, 32, 0, 1], [203329, 32, 203328, 33], [5, 28, 68, 381], [32832, 29, 199627, 2562], [38162, 19, 87215, 202318], [23, 18, 200798, 201355], [177943, 2, 177914, 203343], [162711, 26, 38134, 200491], [39703, 26, 124774, 875]];

sub<GL(2,Integers(203360))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$203360 = 2^{5} \cdot 5 \cdot 31 \cdot 41$$, index $768$, genus $13$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 203329 & 32 \\ 203328 & 33 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 32832 & 29 \\ 199627 & 2562 \end{array}\right),\left(\begin{array}{rr} 38162 & 19 \\ 87215 & 202318 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 200798 & 201355 \end{array}\right),\left(\begin{array}{rr} 177943 & 2 \\ 177914 & 203343 \end{array}\right),\left(\begin{array}{rr} 162711 & 26 \\ 38134 & 200491 \end{array}\right),\left(\begin{array}{rr} 39703 & 26 \\ 124774 & 875 \end{array}\right)$.

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

The torsion field $K:=\Q(E[203360])$ is a degree-$604530907545600000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/203360\Z)$.

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$ Reduction type Serre weight Serre conductor
$2$ split multiplicative $4$ $$31$$
$3$ split multiplicative $4$ $$88970 = 2 \cdot 5 \cdot 7 \cdot 31 \cdot 41$$
$5$ split multiplicative $6$ $$53382 = 2 \cdot 3 \cdot 7 \cdot 31 \cdot 41$$
$7$ nonsplit multiplicative $8$ $$38130 = 2 \cdot 3 \cdot 5 \cdot 31 \cdot 41$$
$31$ nonsplit multiplicative $32$ $$8610 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 41$$
$41$ split multiplicative $42$ $$6510 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 31$$

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4, 8 and 16.
Its isogeny class 266910.ck consists of 8 curves linked by isogenies of degrees dividing 16.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{8}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{31})$$ $$\Z/2\Z \oplus \Z/8\Z$$ not in database $2$ $$\Q(\sqrt{205})$$ $$\Z/16\Z$$ not in database $2$ $$\Q(\sqrt{6355})$$ $$\Z/16\Z$$ not in database $4$ $$\Q(\sqrt{31}, \sqrt{205})$$ $$\Z/2\Z \oplus \Z/16\Z$$ not in database $8$ deg 8 $$\Z/4\Z \oplus \Z/8\Z$$ not in database $8$ deg 8 $$\Z/2\Z \oplus \Z/16\Z$$ not in database $8$ deg 8 $$\Z/32\Z$$ not in database $8$ deg 8 $$\Z/32\Z$$ not in database $8$ deg 8 $$\Z/24\Z$$ not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/16\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/32\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/32\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/24\Z$$ not in database $16$ deg 16 $$\Z/48\Z$$ not in database $16$ deg 16 $$\Z/48\Z$$ 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

No Iwasawa invariant data is available for this curve.

## $p$-adic regulators

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.