# Properties

 Label 637c1 Conductor $637$ Discriminant $-3672178237$ j-invariant $$-\frac{56723625}{13}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy=x^3-x^2-5252x-145223$$ y^2+xy=x^3-x^2-5252x-145223 (homogenize, simplify) $$y^2z+xyz=x^3-x^2z-5252xz^2-145223z^3$$ y^2z+xyz=x^3-x^2z-5252xz^2-145223z^3 (dehomogenize, simplify) $$y^2=x^3-84035x-9378306$$ y^2=x^3-84035x-9378306 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, -1, 0, -5252, -145223])

gp: E = ellinit([1, -1, 0, -5252, -145223])

magma: E := EllipticCurve([1, -1, 0, -5252, -145223]);

oscar: E = EllipticCurve([1, -1, 0, -5252, -145223])

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

 $P$ = $$\left(\frac{4776}{49}, \frac{158761}{343}\right)$$ (4776/49, 158761/343) $\hat{h}(P)$ ≈ $8.8267466432495297542348194658$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Integral points

None

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$637$$ = $7^{2} \cdot 13$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-3672178237$ = $-1 \cdot 7^{10} \cdot 13$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-\frac{56723625}{13}$$ = $-1 \cdot 3^{3} \cdot 5^{3} \cdot 7^{5} \cdot 13^{-1}$ 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: $0.82691117974413006164040154455\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.79468061113529769261405907499\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.2831133310207903\dots$ Szpiro ratio: $5.778926735291447\dots$

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $8.8267466432495297542348194658\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.28040033781113970983160384606\dots$ 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: $1$ ( rounded) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L'(E,1)$ ≈ $2.4750227405405116291414773913$ 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

$\displaystyle 2.475022741 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.280400 \cdot 8.826747 \cdot 1}{1^2} \approx 2.475022741$

# 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

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

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: 420
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$(j)_{-}$
$7$ $1$ $II^{*}$ Additive -1 2 10 0
$13$ $1$ $I_{1}$ 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
$7$ 7B.1.5 7.48.0.6

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 = [[173, 250, 266, 177], [351, 14, 350, 15], [1, 14, 0, 1], [1, 0, 14, 1], [8, 5, 91, 57], [197, 14, 287, 99], [183, 14, 189, 99]]

GL(2,Integers(364)).subgroup(gens)

Gens := [[173, 250, 266, 177], [351, 14, 350, 15], [1, 14, 0, 1], [1, 0, 14, 1], [8, 5, 91, 57], [197, 14, 287, 99], [183, 14, 189, 99]];

sub<GL(2,Integers(364))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$364 = 2^{2} \cdot 7 \cdot 13$$, index $96$, genus $2$, and generators

$\left(\begin{array}{rr} 173 & 250 \\ 266 & 177 \end{array}\right),\left(\begin{array}{rr} 351 & 14 \\ 350 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 197 & 14 \\ 287 & 99 \end{array}\right),\left(\begin{array}{rr} 183 & 14 \\ 189 & 99 \end{array}\right)$.

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

The torsion field $K:=\Q(E[364])$ is a degree-$52835328$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/364\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 7.
Its isogeny class 637c consists of 2 curves linked by isogenies of degree 7.

## Twists

The minimal quadratic twist of this elliptic curve is 637a1, its twist by $-7$.

## 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 $K$ $3$ 3.1.2548.1 $$\Z/2\Z$$ Not in database $6$ 6.0.337599808.2 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $6$ $$\Q(\zeta_{7})$$ $$\Z/7\Z$$ Not in database $8$ 8.2.7348698545643.5 $$\Z/3\Z$$ Not in database $12$ deg 12 $$\Z/4\Z$$ Not in database $18$ 18.0.93862185468913507889152.1 $$\Z/14\Z$$ Not in database $21$ 21.3.26160559119874379648562947375700703.1 $$\Z/7\Z$$ 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 ord ss ss add ord split ord ord ord ord ss ord ss ord ord 1 1,1 1,1 - 1 2 1 1 3 1 1,1 1 1,1 1 1 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

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