Properties

Label 210.a1
Conductor $210$
Discriminant $5670$
j-invariant \( \frac{5763259856089}{5670} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -373, 2623])
 
gp: E = ellinit([1, 1, 0, -373, 2623])
 
magma: E := EllipticCurve([1, 1, 0, -373, 2623]);
 

\(y^2+xy=x^3+x^2-373x+2623\)  Toggle raw display

Mordell-Weil group structure

$\Z\times \Z/{2}\Z$

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

$P$ =  \(\left(13, 7\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.53264658095574456192746745541$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(\frac{43}{4}, -\frac{43}{8}\right) \)  Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(11, -4\right) \), \( \left(11, -7\right) \), \( \left(13, 7\right) \), \( \left(13, -20\right) \), \( \left(31, 133\right) \), \( \left(31, -164\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 210 \)  =  $2 \cdot 3 \cdot 5 \cdot 7$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $5670 $  =  $2 \cdot 3^{4} \cdot 5 \cdot 7 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{5763259856089}{5670} \)  =  $2^{-1} \cdot 3^{-4} \cdot 5^{-1} \cdot 7^{-1} \cdot 17929^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.013583148081496443568986081444\dots$
Stable Faltings height: $0.013583148081496443568986081444\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.53264658095574456192746745541\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $3.5846020263461843307559188552\dots$
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: $ 2 $  = $ 1\cdot2\cdot1\cdot1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $2$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Special value: $ L'(E,1) $ ≈ $ 0.95466300671016443659063754510985106492 $

Modular invariants

Modular form   210.2.a.a

sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 
magma: ModularForm(E);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 64
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is semistable. There are 4 primes of bad reduction:

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$3$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$5$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$7$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

Galois representations

sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

The $\ell$-adic Galois representation has maximal image $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 8.12.0.8

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

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

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit nonsplit nonsplit nonsplit ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 2 1 1 1 1 1 1 1,1 1 1 1 1 3 1 1
$\mu$-invariant(s) 0 0 0 0 0 0 0 0,0 0 0 0 0 0 0 0

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 210.a consists of 3 curves linked by isogenies of degrees dividing 4.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{70}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{35}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{2}) \) \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{2}, \sqrt{35})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.7710244864000000.17 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.8.609892416000000.8 \(\Z/8\Z\) Not in database
$8$ 8.0.416179814400.17 \(\Z/8\Z\) Not in database
$8$ 8.2.4253299470000.4 \(\Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \times \Z/4\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/12\Z\) 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.