Properties

Label 1216.e2
Conductor $1216$
Discriminant $-39845888$
j-invariant \( -\frac{413493625}{152} \)
CM no
Rank $1$
Torsion structure trivial

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -993, 12385])
 
gp: E = ellinit([0, -1, 0, -993, 12385])
 
magma: E := EllipticCurve([0, -1, 0, -993, 12385]);
 

\(y^2=x^3-x^2-993x+12385\)  Toggle raw display

Mordell-Weil group structure

$\Z$

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(9, 64\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.27825869830453934329710178154$

Integral points

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

\((9,\pm 64)\), \((19,\pm 4)\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 1216 \)  =  $2^{6} \cdot 19$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-39845888 $  =  $-1 \cdot 2^{21} \cdot 19 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{413493625}{152} \)  =  $-1 \cdot 2^{-3} \cdot 5^{3} \cdot 19^{-1} \cdot 149^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.42627809312905185368256486534\dots$
Stable Faltings height: $-0.61344267771086611044328331685\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.27825869830453934329710178154\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $2.0053183057830649435951031330\dots$
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: $ 4 $  = $ 2^{2}\cdot1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $1$
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) $ ≈ $ 2.2319890458138393663961927064500415656 $

Modular invariants

Modular form   1216.2.a.e

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);
 

\( q - q^{3} - q^{7} - 2q^{9} + 6q^{11} - 5q^{13} + 3q^{17} - q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 384
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 2 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$(j)_{-}$
$2$ $4$ $I_{11}^{*}$ Additive 1 6 21 3
$19$ $1$ $I_{1}$ 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 for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$3$ 3B 27.36.0.1

$p$-adic regulators

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

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

Iwasawa invariants

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

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3 and 9.
Its isogeny class 1216.e consists of 3 curves linked by isogenies of degrees dividing 9.

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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{2}) \) \(\Z/3\Z\) 2.2.8.1-722.1-b1
$3$ 3.1.152.1 \(\Z/2\Z\) Not in database
$6$ 6.0.3511808.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$6$ 6.0.1801557504.3 \(\Z/3\Z\) Not in database
$6$ 6.6.66724352.1 \(\Z/9\Z\) Not in database
$6$ 6.2.184832.1 \(\Z/6\Z\) Not in database
$12$ 12.2.119973433931988992.9 \(\Z/4\Z\) Not in database
$12$ Deg 12 \(\Z/3\Z \times \Z/3\Z\) Not in database
$12$ 12.0.24904730935296.4 \(\Z/9\Z\) Not in database
$12$ 12.0.12332795428864.1 \(\Z/2\Z \times \Z/6\Z\) Not in database
$18$ 18.0.2110821887190611571419438383104.1 \(\Z/6\Z\) Not in database
$18$ 18.6.107240862022588608007897088.1 \(\Z/18\Z\) 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.